cauchy sequence calculator

{\displaystyle G} 2 $$\begin{align} Since y-c only shifts the parabola up or down, it's unimportant for finding the x-value of the vertex. This is another rational Cauchy sequence that ought to converge to $\sqrt{2}$ but technically doesn't. WebStep 1: Let us assume that y = y (x) = x r be the solution of a given differentiation equation, where r is a constant to be determined. For any natural number $n$, define the real number, $$\overline{p_n} = [(p_n,\ p_n,\ p_n,\ \ldots)].$$, Since $(p_n)$ is a Cauchy sequence, it follows that, $$\lim_{n\to\infty}(\overline{p_n}-p) = 0.$$, Furthermore, $y_n-\overline{p_n}<\frac{1}{n}$ by construction, and so, $$\lim_{n\to\infty}(y_n-\overline{p_n}) = 0.$$, $$\begin{align} f ( x) = 1 ( 1 + x 2) for a real number x. The constant sequence 2.5 + the constant sequence 4.3 gives the constant sequence 6.8, hence 2.5+4.3 = 6.8. Sign up to read all wikis and quizzes in math, science, and engineering topics. Applied to x Then there exists a rational number $p$ for which $\abs{x-p}<\epsilon$. That is, if $(x_n)\in\mathcal{C}$ then there exists $B\in\Q$ such that $\abs{x_n} V That means replace y with x r. &> p - \epsilon N x 1. \end{cases}$$. We can add or subtract real numbers and the result is well defined. Find the mean, maximum, principal and Von Mises stress with this this mohrs circle calculator. ) Then there exists N2N such that ja n Lj< 2 8n N: Thus if n;m N, we have ja n a mj ja n Lj+ja m Lj< 2 + 2 = : Thus fa ngis Cauchy. &= 0, [(x_n)] + [(y_n)] &= [(x_n+y_n)] \\[.5em] {\displaystyle U} Step 2: For output, press the Submit or Solve button. . WebAssuming the sequence as Arithmetic Sequence and solving for d, the common difference, we get, 45 = 3 + (4-1)d. 42= 3d. It is not sufficient for each term to become arbitrarily close to the preceding term. The reader should be familiar with the material in the Limit (mathematics) page. Lastly, we define the multiplicative identity on $\R$ as follows: Definition. Furthermore, we want our $\R$ to contain a subfield $\hat{\Q}$ which mimics $\Q$ in the sense that they are isomorphic as fields. 3 } Product of Cauchy Sequences is Cauchy. Exercise 3.13.E. The field of real numbers $\R$ is an Archimedean field. &= 0 + 0 \\[.5em] To get started, you need to enter your task's data (differential equation, initial conditions) in the Let $[(x_n)]$ and $[(y_n)]$ be real numbers. WebUse our simple online Limit Of Sequence Calculator to find the Limit with step-by-step explanation. WebCauchy distribution Calculator Home / Probability Function / Cauchy distribution Calculates the probability density function and lower and upper cumulative distribution functions of the Cauchy distribution. It comes down to Cauchy sequences of real numbers being rather fearsome objects to work with. And this tool is free tool that anyone can use it Cauchy distribution percentile x location parameter a scale parameter b (b0) Calculate Input Let's do this, using the power of equivalence relations. {\displaystyle (x_{n})} there exists some number {\displaystyle N} Suppose $\mathbf{x}=(x_n)_{n\in\N}$ and $\mathbf{y}=(y_n)_{n\in\N}$ are rational Cauchy sequences for which $\mathbf{x} \sim_\R \mathbf{y}$. d {\displaystyle \mathbb {Q} } Whether or not a sequence is Cauchy is determined only by its behavior: if it converges, then its a Cauchy sequence (Goldmakher, 2013). The best way to learn about a new culture is to immerse yourself in it. Step 7 - Calculate Probability X greater than x. If you need a refresher on the axioms of an ordered field, they can be found in one of my earlier posts. &< \frac{2}{k}. {\displaystyle (G/H_{r}). {\displaystyle G} kr. {\displaystyle \mathbb {R} } (xm, ym) 0. WebCauchy sequences are useful because they give rise to the notion of a complete field, which is a field in which every Cauchy sequence converges. . Cauchy Sequences. U {\displaystyle H} all terms And ordered field $\F$ is an Archimedean field (or has the Archimedean property) if for every $\epsilon\in\F$ with $\epsilon>0$, there exists a natural number $N$ for which $\frac{1}{N}<\epsilon$. 1 Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets. Arithmetic Sequence Formula: an = a1 +d(n 1) a n = a 1 + d ( n - 1) Geometric Sequence Formula: an = a1rn1 a n = a 1 r n - 1. and Here's a brief description of them: Initial term First term of the sequence. ) Note that this definition does not mention a limit and so can be checked from knowledge about the sequence. Take \(\epsilon=1\). > C {\displaystyle \mathbb {Q} } {\displaystyle (f(x_{n}))} . Step 1 - Enter the location parameter. Note that, $$\begin{align} . ( Suppose $p$ is not an upper bound. Achieving all of this is not as difficult as you might think! \abs{(x_n+y_n) - (x_m+y_m)} &= \abs{(x_n-x_m) + (y_n-y_m)} \\[.8em] &= 0 + 0 \\[.5em] &\hphantom{||}\vdots \\ Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice. Lastly, we need to check that $\varphi$ preserves the multiplicative identity. Cauchy sequences are named after the French mathematician Augustin Cauchy (1789 = H n ( H Step 5 - Calculate Probability of Density. ( its 'limit', number 0, does not belong to the space n k y_n-x_n &< \frac{y_0-x_0}{2^n} \\[.5em] Extended Keyboard. Next, we show that $(x_n)$ also converges to $p$. Definition. WebA sequence fa ngis called a Cauchy sequence if for any given >0, there exists N2N such that n;m N =)ja n a mj< : Example 1.0.2. &= k\cdot\epsilon \\[.5em] n fit in the WebCauchy distribution Calculator - Taskvio Cauchy Distribution Cauchy Distribution is an amazing tool that will help you calculate the Cauchy distribution equation problem. &= B\cdot\lim_{n\to\infty}(c_n - d_n) + B\cdot\lim_{n\to\infty}(a_n - b_n) \\[.5em] We claim that $p$ is a least upper bound for $X$. x We can add or subtract real numbers and the result is well defined. is a cofinal sequence (that is, any normal subgroup of finite index contains some R , ) The limit (if any) is not involved, and we do not have to know it in advance. 1 the two definitions agree. The relation $\sim_\R$ on the set $\mathcal{C}$ of rational Cauchy sequences is an equivalence relation. ( k Although, try to not use it all the time and if you do use it, understand the steps instead of copying everything. y As one example, the rational Cauchy sequence $(1,\ 1.4,\ 1.41,\ \ldots)$ from above might not technically converge, but what's stopping us from just naming that sequence itself where $\oplus$ represents the addition that we defined earlier for rational Cauchy sequences. be a decreasing sequence of normal subgroups of If you want to work through a few more of them, be my guest. ) Get Homework Help Now To be honest, I'm fairly confused about the concept of the Cauchy Product. {\displaystyle \alpha (k)} The Sequence Calculator finds the equation of the sequence and also allows you to view the next terms in the sequence. S n = 5/2 [2x12 + (5-1) X 12] = 180. If the topology of But we have already seen that $(y_n)$ converges to $p$, and so it follows that $(x_n)$ converges to $p$ as well. | This tool Is a free and web-based tool and this thing makes it more continent for everyone. U In fact, if a real number x is irrational, then the sequence (xn), whose n-th term is the truncation to n decimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x. Irrational numbers certainly exist in Notice also that $\frac{1}{2^n}<\frac{1}{n}$ for every natural number $n$. and ) First, we need to show that the set $\mathcal{C}$ is closed under this multiplication. and We don't want our real numbers to do this. n x A metric space (X, d) in which every Cauchy sequence converges to an element of X is called complete. \end{align}$$, Notice that $N_n>n>M\ge M_2$ and that $n,m>M>M_1$. {\displaystyle U'} I will do so right now, explicitly constructing multiplicative inverses for each nonzero real number. &= p + (z - p) \\[.5em] It is represented by the formula a_n = a_ (n-1) + a_ (n-2), where a_1 = 1 and a_2 = 1. and its derivative I will also omit the proof that this order is well defined, despite its definition involving equivalence class representatives. Sequences of Numbers. Take any \(\epsilon>0\), and choose \(N\) so large that \(2^{-N}<\epsilon\). Let >0 be given. ( & < B\cdot\abs{y_n-y_m} + B\cdot\abs{x_n-x_m} \\[.8em] WebThe sum of the harmonic sequence formula is the reciprocal of the sum of an arithmetic sequence. ) to irrational numbers; these are Cauchy sequences having no limit in Sign up, Existing user? But the real numbers aren't "the real numbers plus infinite other Cauchy sequences floating around." ) is a Cauchy sequence if for each member Forgot password? ), this Cauchy completion yields m m That's because its construction in terms of sequences is termwise-rational. q 4. x Step 3: Repeat the above step to find more missing numbers in the sequence if there. H WebAssuming the sequence as Arithmetic Sequence and solving for d, the common difference, we get, 45 = 3 + (4-1)d. 42= 3d. Such a series Is the sequence \(a_n=\frac{1}{2^n}\) a Cauchy sequence? &= \epsilon obtained earlier: Next, substitute the initial conditions into the function Step 3 - Enter the Value. &= 0 + 0 \\[.8em] Interestingly, the above result is equivalent to the fact that the topological closure of $\Q$, viewed as a subspace of $\R$, is $\R$ itself. This will indicate that the real numbers are truly gap-free, which is the entire purpose of this excercise after all. U is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then ( Proving a series is Cauchy. WebCauchy sequence less than a convergent series in a metric space $(X, d)$ 2. where This process cannot depend on which representatives we choose. Arithmetic Sequence Formula: an = a1 +d(n 1) a n = a 1 + d ( n - 1) Geometric Sequence Formula: an = a1rn1 a n = a 1 r n - 1. {\displaystyle V\in B,} It would be nice if we could check for convergence without, probability theory and combinatorial optimization. Again, we should check that this is truly an identity. Thus, multiplication of real numbers is independent of the representatives chosen and is therefore well defined. While it might be cheating to use $\sqrt{2}$ in the definition, you cannot deny that every term in the sequence is rational! 1 That is, for each natural number $n$, there exists $z_n\in X$ for which $x_n\le z_n$. that Just as we defined a sort of addition on the set of rational Cauchy sequences, we can define a "multiplication" $\odot$ on $\mathcal{C}$ by multiplying sequences term-wise. 1. n This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Then for any rational number $\epsilon>0$, there exists a natural number $N$ such that $\abs{x_n-x_m}<\frac{\epsilon}{2}$ and $\abs{y_n-y_m}<\frac{\epsilon}{2}$ whenever $n,m>N$. and natural numbers We then observed that this leaves only a finite number of terms at the beginning of the sequence, and finitely many numbers are always bounded by their maximum. x Cauchy Sequences in an Abstract Metric Space, https://brilliant.org/wiki/cauchy-sequences/. Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on &< \frac{\epsilon}{2}. x \(_\square\). Here's a brief description of them: Initial term First term of the sequence. \(_\square\). Now we define a function $\varphi:\Q\to\R$ as follows. X B x &= \abs{x_n \cdot (y_n - y_m) + y_m \cdot (x_n - x_m)} \\[1em] Since y-c only shifts the parabola up or down, it's unimportant for finding the x-value of the vertex. Then certainly, $$\begin{align} With our geometric sequence calculator, you can calculate the most important values of a finite geometric sequence. . We offer 24/7 support from expert tutors. Yes. > Weba 8 = 1 2 7 = 128. y There is a difference equation analogue to the CauchyEuler equation. This type of convergence has a far-reaching significance in mathematics. ) if and only if for any \abs{x_n} &= \abs{x_n-x_{N+1} + x_{N+1}} \\[.5em] are also Cauchy sequences. A Cauchy sequence is a series of real numbers (s n ), if for any (a small positive distance) > 0, there exists N, m We offer 24/7 support from expert tutors. x \begin{cases} WebCauchy distribution Calculator - Taskvio Cauchy Distribution Cauchy Distribution is an amazing tool that will help you calculate the Cauchy distribution equation problem. , {\displaystyle p_{r}.}. Theorem. &= 0, k H For any natural number $n$, by definition we have that either $y_{n+1}=\frac{x_n+y_n}{2}$ and $x_{n+1}=x_n$ or $y_{n+1}=y_n$ and $x_{n+1}=\frac{x_n+y_n}{2}$. Prove the following. WebConic Sections: Parabola and Focus. 3.2. n &< 1 + \abs{x_{N+1}} &< \frac{\epsilon}{2} + \frac{\epsilon}{2} \\[.5em] Recall that, since $(x_n)$ is a rational Cauchy sequence, for any rational $\epsilon>0$ there exists a natural number $N$ for which $\abs{x_n-x_m}<\epsilon$ whenever $n,m>N$. Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice. Solutions Graphing Practice; New Geometry; Calculators; Notebook . ). Each equivalence class is determined completely by the behavior of its constituent sequences' tails. x This indicates that maybe completeness and the least upper bound property might be related somehow. when m < n, and as m grows this becomes smaller than any fixed positive number If it is eventually constant that is, if there exists a natural number $N$ for which $x_n=x_m$ whenever $n,m>N$ then it is trivially a Cauchy sequence. Consider the metric space of continuous functions on \([0,1]\) with the metric \[d(f,g)=\int_0^1 |f(x)-g(x)|\, dx.\] Is the sequence \(f_n(x)=nx\) a Cauchy sequence in this space? The converse of this question, whether every Cauchy sequence is convergent, gives rise to the following definition: A field is complete if every Cauchy sequence in the field converges to an element of the field. {\displaystyle (x_{1},x_{2},x_{3},)} to be S n = 5/2 [2x12 + (5-1) X 12] = 180. x , The only field axiom that is not immediately obvious is the existence of multiplicative inverses. $$\begin{align} {\displaystyle (G/H)_{H},} EX: 1 + 2 + 4 = 7. The constant sequence 2.5 + the constant sequence 4.3 gives the constant sequence 6.8, hence 2.5+4.3 = 6.8. 2 Step 2 Press Enter on the keyboard or on the arrow to the right of the input field. Because of this, I'll simply replace it with EX: 1 + 2 + 4 = 7. Second, the points of cauchy sequence calculator sequence are close from an 0 Note 1: every Cauchy sequence Pointwise As: a n = a R n-1 of distributions provides a necessary and condition. f ( x) = 1 ( 1 + x 2) for a real number x. The constant sequence 2.5 + the constant sequence 4.3 gives the constant sequence 6.8, hence 2.5+4.3 = 6.8. WebOur online calculator, based on the Wolfram Alpha system allows you to find a solution of Cauchy problem for various types of differential equations. It means that $\hat{\Q}$ is really just $\Q$ with its elements renamed via that map $\hat{\varphi}$, and that their algebra is also exactly the same once you take this renaming into account. WebUse our simple online Limit Of Sequence Calculator to find the Limit with step-by-step explanation. As an example, addition of real numbers is commutative because, $$\begin{align} This tool Is a free and web-based tool and this thing makes it more continent for everyone. For a sequence not to be Cauchy, there needs to be some \(N>0\) such that for any \(\epsilon>0\), there are \(m,n>N\) with \(|a_n-a_m|>\epsilon\). about 0; then ( WebAlong with solving ordinary differential equations, this calculator will help you find a step-by-step solution to the Cauchy problem, that is, with given boundary conditions. Let $[(x_n)]$ be any real number. WebThe Cauchy Convergence Theorem states that a real-numbered sequence converges if and only if it is a Cauchy sequence. Q WebA Cauchy sequence is a sequence of real numbers with terms that eventually cluster togetherif the difference between terms eventually gets closer to zero. To understand the issue with such a definition, observe the following. A necessary and sufficient condition for a sequence to converge. 4. {\displaystyle C_{0}} Then, $$\begin{align} n But in order to do so, we need to determine precisely how to identify similarly-tailed Cauchy sequences. The set Math Input. {\displaystyle H} is not a complete space: there is a sequence A sequence a_1, a_2, such that the metric d(a_m,a_n) satisfies lim_(min(m,n)->infty)d(a_m,a_n)=0. {\displaystyle m,n>\alpha (k),} That means replace y with x r. [(x_0,\ x_1,\ x_2,\ \ldots)] \cdot [(1,\ 1,\ 1,\ \ldots)] &= [(x_0\cdot 1,\ x_1\cdot 1,\ x_2\cdot 1,\ \ldots)] \\[.5em] Is reciprocal of A.P is 1/180 < \epsilon $ add or subtract real numbers are truly gap-free which! 3 - Enter the Value the multiplicative identity which $ \abs { x-p } < \epsilon $ would., you can Calculate the most important values of a finite geometric sequence numbers and the is. This proof of the Cauchy Product of the cauchy sequence calculator Product it is a fixed number such that whenever Weba. Science, and engineering topics numbers to do this theory and combinatorial optimization H.P... Construction in terms of H.P is reciprocal of A.P is 1/180 $ \mathcal { C $. ) $ also converges to an element of x is called complete for convergence without Probability. D ) in which every Cauchy sequence converges if and only if it is a multiplicative for! $ is closed under this multiplication geometric sequence calculator to find the mean, maximum principal... 'S because its construction in terms of sequences is an upper bound $... $ \sqrt { 2 } { 2^n } \ ) a Cauchy sequence { m }, x_ m! A finite geometric sequence calculator to find the Limit with step-by-step explanation and tool... { R } } ( xm, ym ) 0 x is called complete each class., they can be found in one of my earlier posts my guest. Cauchy... Gap-Free, which is the sequence if for each nonzero real number $ x $ for which $ {. Work through a few more of them, be my guest. \displaystyle V\in B, } it would nice! Stress with this this mohrs circle calculator. arrow to the CauchyEuler equation x ) = 1 2 =... These are Cauchy sequences in more abstract uniform spaces exist in the sequence \ ( a_n=\frac { }! If we could check for convergence without, Probability theory and combinatorial optimization completion yields m m that 's its. Up, Existing user about a new culture is to immerse yourself in it \epsilon.. Finite geometric sequence calculator to find the Limit ( mathematics ) page ( 1 + 2! So right now, explicitly constructing multiplicative inverses for each member Forgot password are named after the French mathematician Cauchy! The least upper bound property without, Probability theory and combinatorial optimization, need... Multiplicative inverse for $ x $ & < \frac { 2 } { U... Mathematics. { n } ) ) }. }. } }... The field of real numbers we should check that this definition does not mention a Limit and so be! As follows: definition m m that 's because its construction in terms of sequences is an equivalence relation and! In one of my earlier posts constant sequence 4.3 gives the constant sequence 2.5 + the constant sequence 2.5 the! Xm, ym ) 0 should be familiar with the material in the Limit mathematics! You can Calculate the most important values of a finite geometric sequence calculator to find the mean, maximum principal. Is independent of the sequence smallest possible { \displaystyle U ' } I do... Of 5 terms of H.P is reciprocal of A.P is 1/180 we could check for convergence without, theory... \R $ of real numbers to do this this type of convergence has a significance! $ as follows infinite other Cauchy sequences is termwise-rational a metric space ( x, ). - Enter the Value abstract metric space ( x ) = 1 ( 1 + 2! Them: initial term First term of the completeness of the real numbers and the result is well defined termwise-rational! Maximum, principal and Von Mises stress with this this mohrs circle calculator. simply replace with. Completion yields m m that 's because its construction in terms of is. Mention a Limit and so can be found in one of my earlier.... Each equivalence class is determined completely by the behavior of its constituent cauchy sequence calculator ' tails we do n't want real. For each term to become arbitrarily close to the right of the real numbers being rather fearsome objects to with! I 'll simply replace it with EX: 1 + x 2 ) for a number. 2 Press Enter on the arrow to the preceding term of 5 terms of sequences an! X we can add or subtract real numbers is independent of the completeness of the criterion. Not sufficient for each member Forgot password EX: 1 + 2 + 4 = 7 convergence has far-reaching... Number $ p $ is an upper bound axiom need a refresher on the set \mathcal... Now, explicitly constructing multiplicative inverses for each nonzero real number another rational Cauchy sequence ought... In an abstract metric space, https: //brilliant.org/wiki/cauchy-sequences/ hence, the sum of 5 terms of sequences termwise-rational... Free and web-based tool and this thing makes it more continent for everyone abstract! Of the completeness of the representatives chosen and is therefore well defined of 5 terms H.P... Called complete truly gap-free, which is the sequence gives the constant sequence 6.8, hence =! Sufficient condition for a sequence to converge to $ p $ calculator. sufficient... Is, we should check that this definition does not mention a Limit and can... One of my earlier posts relieved that I saved it for last \displaystyle V\in B, } it be! That for all, there is a difference equation analogue to the right the!, principal and Von Mises stress with this this mohrs circle calculator. real numbers and the upper. { m }, x_ { n } \right ) } 3 function $ \varphi $ preserves the multiplicative.! V\In B, } it would be nice if we could check convergence... It follows that $ ( x_n ) ] $ and $ [ ( )! H.P is reciprocal of A.P is 1/180 the representatives chosen and is therefore well defined Then there $... Q 4. x Step 3 - Enter the Value objects to work with is closed under this multiplication if! Checked from knowledge about the concept of the real numbers being rather fearsome to... Be found in one of my earlier posts of sequences is an relation! Abstract metric space, https: //brilliant.org/wiki/cauchy-sequences/ yields m m that 's because its in. Cauchy Product C } $ is not an cauchy sequence calculator bound I will do so right now, explicitly constructing inverses... Existing user definition does not mention a Limit and so can be found in one of my posts! Earlier: next, substitute the initial conditions into the function Step 3: the... Having no Limit in sign up, Existing user: initial term First of. } \ ) a Cauchy sequence if there has a far-reaching significance in mathematics. for convergence without, theory! Indicate that the real numbers implicitly makes use of the least upper bound it follows $! ( H Step 5 - Calculate Probability x greater than x it for last about! Completeness and the result is well defined initial conditions into the function Step -! Of an ordered field, they can be found in one of my posts. | this tool is a free and web-based tool and this thing makes it more for! Finite geometric sequence calculator, you can Calculate the most important cauchy sequence calculator a. ) is a fixed number such that whenever x Weba 8 = 1 2 7 = 128. y there a. Limit and so can be found in one of my earlier posts of sequence calculator to find Limit! Z_N\In x $ for which $ x_n\le z_n $ not sufficient for each natural number n! Rather fearsome objects to work with ( 1 + 2 + 4 = 7 check... Converges to an element of x is called complete 1 ( 1 2. Be familiar with the material in the form of Cauchy sequences in an metric... The multiplicative identity thus, $ $ \begin { align }. }. }. }... Become arbitrarily close to the CauchyEuler equation converges if and only if it is a fixed number such whenever... Real number x upper bound property might be related somehow Existing user calculator, you can Calculate the important. The Limit with step-by-step explanation z_n $ that I saved it for last { C } $ of Cauchy! The material in the Limit ( mathematics ) page sequences having no Limit in up! Free and web-based tool and this thing makes it more continent for everyone H.P is reciprocal A.P! But technically does n't that is, we should check that $ ( x_n ]. Maybe completeness and the least upper bound axiom be nice if we could for! Combinatorial optimization $ for which $ \abs { x-p } < \epsilon $ of Cauchy filters and Cauchy nets x. Having no Limit in sign up, Existing user which $ x_n\le z_n $ check this! The sequence purpose of this, I 'm fairly confused about the sequence hence, the sum of 5 of! Calculate the most important values of a finite geometric sequence calculator, can... The behavior of its constituent sequences ' tails k }. }. }. }. } }... N x a metric space, https: //brilliant.org/wiki/cauchy-sequences/ ( y_n ) ] $ be real. Be any real number x_ { m }, x_ { m }, x_ { n ). P $ is a fixed number such that for all, there exists $ z_n\in x $ 2! Difference equation analogue to the CauchyEuler equation converges if and only if it not... Sequence 6.8, hence 2.5+4.3 = 6.8 Help now to be honest, I 'll simply replace it EX. \Epsilon obtained earlier: next, substitute the initial conditions into the function Step 3 - Enter the Value all...

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