Questions tagged [alternating-expression]
For questions related to alternating expression (or series). It is a sequence, whose terms change sign (i.e. if a term $a_n$ is positive then $a_{n+1}$ is negative and vice versa).
57 questions
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4
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Combinatorial sum with alternating signs.
Compute $$\sum_{n=0}^{10}(-1)^n\binom{10}{n}\binom{12+n}{n}.$$
How do I calculate it using well known identities?I've tried more or less every identity I know with alternating signs but I end up with ...
- 175
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vote
0
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49
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Formula for an alternating sum over weight $ w $ bit strings
Let $ b_0 $ be a length $ n $ bit string of Hamming weight $ w_0 $. The sum
$$
\sum_{wt(b)=w} (-1)^{b ~\cdot~ b_0}
$$
over all length $ n $ bit strings $ b $ of some fixed Hamming weight $ w $ has $ \...
- 9,194
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vote
0
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Sum of alternating binomial coefficients that is evaluated on a finite subset of positive integers
Let $n\in\mathbb{N}$ and $I\subset\mathbb{N}\cup\{0\}$.
For simplicity, I define
$$S_n(I) := \sum_{i \in I}(-1)^i\binom{n}{i}.$$
For $n\in\mathbb{N}$, it is well-known that $S_n(\{0,1,\dots,n\})=0$. ...
- 91
2
votes
1
answer
104
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How to prove a formula on sum of products of alternating sign Vandermonde convolution?
The alternating sign Vandermonde convolution refers to the following sum
\begin{align}
V(r,s,N) = \sum_{k=0}^N (-1)^k \tbinom{r}{k}\tbinom{s}{N-k}.
\end{align}
Let $\Delta^{\pm} (m_0,m_1) := |m_0\pm ...
- 75
2
votes
1
answer
104
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Sum of alternating binomial coefficients for $n=6k$ and/or $n=6k+4$ [closed, rewritten in other question]
I have a simple problem, but I don't know a way to prove (or disprove) it.
I wonder if this problem is known already or if the solution can be found somewhere. Does anyone have any idea on how to ...
- 91
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votes
1
answer
47
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Exact value of $\lim_{x\to\infty}\frac{\sqrt2+1}{x\left(8-16\sum_{n=1}^x\frac1{64n^2-1}-\pi\sqrt2-\pi\right)}$, as derived from simple series for pi.
I was reading Euler's E041 ("Concerning the sums of series of reciprocals.") and found some sequences for $\pi$. I believe his derivations are not properly justified. I combined
$$\sqrt2\...
- 384
2
votes
2
answers
131
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A finite sum involving alternate inverse binomial coefficients
$ \sum_{r=1}^{2023} (\frac{(-1)^{r-1}r)}{\mathrm{C}_{r}^{2024}}) $
My approach was to add up $1$st and $2023$rd term, $2nd$ and $2022$nd, $3$rd and $2021$st and so on
$T_r + T_{2024-r}= \frac{(2024(-...
- 35
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vote
0
answers
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How to prove for torsion free connections $A(\nabla^2 w)=0$
It's known that exterior derivative of an $n$-form $w$ can be expressed by any torsion free connection on tensor fields of a manifold as$$dw(X_0,\cdots,X_p)=\sum_i(-1)^i(\nabla_{X_i}w)(X_1,\cdots,\hat{...
- 1,194
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votes
1
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49
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Finding the interval and radius of convergence of a power series with an alternating term
The series in question is:
$$
\sum_{n = 0}\left[2 + \left(-1^n\right)\right]x^{n - 1}
$$
I tried applying ratio test: However, I will still end up with alternating ...
- 61
3
votes
1
answer
222
views
An interesting alternating sum $\sum _{k=0}^n\left(-1\right)^k\binom{k}{m}$
i was messing around with some combinatorial type sums and started thinking about a certain alternate sum that is the following : for $(n,m) \in \mathbb{N}^2$ what would be the value of :
$$S_{m,n}=\...
- 407
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0
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50
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Combinatorial identity (lower summation) [duplicate]
How to prove $${n \choose m}-{n \choose m+1} +{n \choose m+2}-\dots+{n \choose n}=
{n-1 \choose m-1}$$
I tried proving this using binomial expansion adding and subtracting the first (m-1) terms but ...
- 1
2
votes
1
answer
68
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Equivalence of the two functors $Alt^{k}(-*)$ and $(Alt^{k})^{*}$
I am finishing Vector Analysis of Klaus Jänich. I am stuck at chapter $12$ because I am confused about a notation. I hope some of you could untangle it for me.
Lemma
We can interpret each $\varphi \in ...
- 337
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1
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104
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Evaluating $\lim_{n\to\infty} \frac{(-1)^{n}n^{2n}}{(2n)!}$
I have tried substituting $(2n)! \sim \sqrt{4\pi n} \left(\frac{2n}{e}\right)^{2n}$ from Stirling's approximation:
$$\begin{align*}
\lim_{n\to\infty} \frac{(-1)^{n}n^{2n}}{(2n)!} &= \lim_{n\to\...
1
vote
1
answer
49
views
These constrained alternating series always satisfy an inequality.
Let $x_0 \in \Bbb{N}$ and suppose that $x_1 \lt \frac{x_0}{2}$, while $0 = x_n \leq \dots \leq x_3 \leq x_2\leq x_1$.
Then is it possible that:
$$
\sum_{i = 0}^n (-1)^i x_i \gt 1
$$
no matter what ...
- 21.6k
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vote
1
answer
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Identity involving alternating multinomial terms
Take $x_{1},\ldots,x_{k + 1}$ to be distinct real numbers such that, with $k \geq 1$ some positive integer.
I want to show the following:
$$
0 = -\sum_{1\ \leq\ i\ \leq\ k + 1}x_{i}^{k} +
\sum_{1\ \...
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0
answers
105
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Why this does not add up to 0?
i was wondering, why the following sum does not add up to $0$. Consider the following sum of $S_n$ :
$$\sum_{n=0}^\infty S_n = S_1 + S_2 + S_3 + ... = \epsilon$$
And the specific elements looks like ...
- 23
1
vote
3
answers
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Combinatorical identity $\sum_{k = d-i}^d (-1)^{k-d+i} \binom{k}{d-i} \cdot \binom{n}{d-k} = \binom{n-d+i-1}{i}$
How do we proove the following ugly identity of binomial coefficients?
$$\sum_{k = d-i}^d (-1)^{k-d+i} \binom{k}{d-i} \cdot \binom{n}{d-k} = \binom{n-d+i-1}{i}$$
First I thought we could use ...
- 141
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2
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Asymptotic Behaviour of the Remainder of Certain Alternating Series
Let $a,b >0$ be real constants. Empirical observation (as in: asking WolframAlpha) suggests
$$ \lim_{n\to \infty} n \cdot \sum_{k=0}^\infty (\frac{1}{n+ak} - \frac{1}{n+b+ak}) = \frac{b}{a} \tag{$...
- 27.8k
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votes
0
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Request bibliographic reference(s) for finite alternating sums with Eulerian numbers
I would like to know a bibliographic reference for a math formula. I found this formula on Wikipedia, but no reference is given.
It's the 2nd formula ($A(n,k)$ is an Eulerian number):
$$\sum_{k=0}^{n-...
- 27
0
votes
0
answers
51
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Calculating the number of terms required to approximate total sum within a specific error
I have the following infinite series. I am supposed to calculate the number of terms required to approximate the total sum of the series with an error of less than 0.01.
$$
\sum_{k=1}^{\infty}{(-1)^k\...
- 23
10
votes
2
answers
230
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Alternating sum of reciprocals of binomial coefficients
I'm looking for a simple proof of the identity
$$ \sum_{k=0}^n \frac{(-1)^k}{\binom{n}{k}} = \frac{n+1}{n+2} (1+(-1)^n) $$
relying only on elementary properties of binomial coefficients. I obtained ...
- 1,773
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votes
2
answers
144
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Why is a=43 an exception?
There is a sequence. The first integer is positive, the second integer is negative. An alternating part is a part that switches between positive and negative (0 is not included)
(a) I have a complete ...
- 9
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0
answers
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$\sum_{j=0}^\infty(-1)^j\frac{1 -e^{(aj^2 + b)M}}{1-e^{aj^2 +b}}$?
I've recently come across the following series, and I am trying to figure out if there is a closed form solution, or an asymptotic behavior for large M:
$$
\sum_{j=0}^\infty(-1)^j\frac{1 -e^{(aj^2 + b)...
- 53
1
vote
1
answer
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What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?
Let all untyped variables be natural numbers.
Formula?
Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$.
What is the 2-case closed form for $\sum_{i = 1}^{x} \...
- 21.6k
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1
answer
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On the fractional parts of the roots of the Alternating Harmonic Numbers
We define $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-\ln2\right)$$As the $x$th Alternating Harmonic Number (test out a few values to see why). Let $x_n$ be the $n$th ...
- 6,707
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49
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What does it mean, The space $L_n (V^n;K)$ of alternating n-linear forms is of dimension one?
Reading about the fundamental theorem of alternating applications which says
Given 2 vector spaces over $K$, $(V;K)$ and $(W;K)$. If $dim\ \ V=n$ and a base of V is {$u_1...u_n$}
I saw that there is ...
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1
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How to deduce: $T(x_1, \dots , x_i+x_j, \dots, x_i+x_j, \dots , x_n) = 0$?
studying about alternating multilinear applications I came across this expression. I understand that it represents the demonstration that an application T is antisymmetric (since it changes sign when ...
- 29
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0
answers
70
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Property of the alternating operator
Let $\Lambda^kV^*$ be the space of alternating k-linear forms on $V^k:=V \times \dots \times V$.
Such that for $\omega \in \Lambda^kV^*$,
$$\omega(v_1, \dots, v_i+v_j,\dots, v_k)=\omega(v_1, \dots, ...
1
vote
0
answers
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alternate series convergence
I am studying calculus, and now I got some practice from my lecture
which is as follows.
Decide whether the series $\sum_{n=1}^{\infty}\frac{({-1})^n}{(n-1)}$ is convergent, divergent, absolute ...
- 11
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votes
0
answers
90
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An identity involving alternative sum of triple product of binomial coefficients
I can not motivative the problem to be honest, but the following identity has shown up in my research. I have verified the identity for large enough values of $k$ using sagemath. Now, I want to prove ...
- 312
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votes
3
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How do you find the value of $\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n^2}$? [closed]
Extra information which may be useful is that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ equals $\frac{\pi^2}{6}$ (Euler's solution to the Basel Problem).
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Rearrange the terms in the alternating harmonic series such that the sum is $s \in \mathbb{R}$
I have the following problem.
Rearrange the terms in the alternating harmonic series such that the sum is $s \in \mathbb{R}$
I thought about proving it with Riemann's rearrangement theorem, but I am ...
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A tight positivity conjecture about sums over divisors of square-free integers.
Let $p_n$ be the $n$th prime number and all variables, unless otherwise specified, are natural numbers.
Conjecture:
For all square-free $n \geq 2$, the following function evaluates to a positive ...
- 21.6k
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1
answer
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The function $f(n) = \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor$ has no fixed point $f(n) = n$?
Definition.
$$
f(n) := \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor
$$
where $p_{n+1}$ is the $(n+1)$th prime number. And where it is understood that each ...
- 21.6k
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Find an alternating cycle in a graph with perfect matching.
Given a graph $G(V,E)$, we consider that there is a perfect matching $M$ in $G$. The edges in $M$ is red and the edges not in $M$ is blue. So is there a polynomial time algorithm that can find a red-...
4
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2
answers
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Evaluating $ \sum\limits_{n=1}^{N-1} \frac{(-1)^n}{\sqrt{1-\cos{\frac{2\pi n}{N}}}} $
I have come across this sum:
$$\sum\limits_{n=1}^{N-1} \frac{(-1)^n}{\sqrt{1-\cos{\frac{2\pi n}{N}}}} $$
where $N$ is an even integer (it evaluates to $0$ for odd $N$). How does one evaluate this sum (...
- 132
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0
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45
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Simplifying $\cos(\frac{2\pi n}{3}), n \in \mathbb{Z}$ to an alternating series
I'm trying to simplify the function $\cos\left(\frac{2\pi n}{3}\right)$ for integer $n$ into some sort of alternating series expression. For reference, I know that $\cos(\pi n)$ can be written as:
$$
\...
- 37
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votes
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answers
40
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This alternating sum of fractional floor functions over the divisors of primorial is always a non-decreasing function (the general case).
Define the family of functions for $n \geq 1$.
$$
f_n(x) = \sum_{d \mid p_n\#}(-1)^{\omega(d)}\sum_{0 \leq r \lt d \\ r^2 = 1 \pmod d}\left\lfloor \frac{x - r}{d}\right\rfloor
$$
Conjecture. In ...
- 21.6k
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1
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How can we prove that this alternating summation involving fractional floor functions is non-decreasing?
Question. How can we prove that this function is non-decreasing?
That is:
$$
f: \Bbb{R} \to \Bbb{Z} \\
f(x) = [\frac{x}{1}] - [\frac{x-1}{2}] - [\frac{x - 1}{3}] - [\frac{x - 2}{3}] - [\frac{x - 1}{...
- 21.6k
5
votes
1
answer
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If $(a_n)$ is a decreasing real sequence and $\sum a_n$ converges, then does $\sum (-1)^n n a_n\ $ converge?
"Motivation"/Introduction:
If $(a_n)$ is a decreasing real sequence and $\displaystyle\sum a_n $ converges, then $n a_n \to 0,\ $ for example, by the Cauchy Condensation test.
If $(a_n)$ ...
- 22.6k
0
votes
0
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111
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$t(x) - t(x/2) + t(x/3) - t(x/4) + ... = 0$ implies $t(x) + t(x/2)+t(x/3)+t(x/4) = C x$?
Inspired by this mysterious function :
$f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$ and $\lim_{n \to \infty} \frac{f(n)}{\pi(n)} = 1$?
I started to wonder since the alternating sum equals zero :
$$f(x) ...
- 17.1k
3
votes
0
answers
240
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Two identities involving binomial coefficients and double factorials
I'm trying to prove the following identities:
$$\forall t\geqslant2,\forall 0\leqslant i\leqslant\left\lfloor\frac{t}{2}\right\rfloor,S_{\text{even}}(i, t)=\sum_{k=0}^{\left\lfloor\frac{t}{2}\right\...
- 209
1
vote
0
answers
92
views
Interchanging summations over sets
In order to understand two different definitions of the same function $f$ (as stated by Besner, 2022), I am trying to prove that those expressions are equal:
$\Delta(A) = f(A) - \sum_{B \subset A} \...
- 43
3
votes
1
answer
230
views
A curious limit for $-\frac{1}{2}$
How to prove this ?
$$-\frac{1}{2} = \lim_{x\to+\infty}\sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \sqrt{\ln 2n}}$$
It reminded me of the fact that
$$-\frac\pi2 = \lim_{x\to+\infty}\sum_{n=1}^{\...
- 17.1k
-2
votes
1
answer
58
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Harmonic series and its alternating counterpart
I can't understand intuitively why the series
$$ \sum^{\infty}_{k=1} \frac{1}{k} $$
diverges while its counterpart with only the alternating signs does the opposite (converges)
$$\sum^{\infty}_{k=1} \...
- 315
0
votes
1
answer
37
views
Not all black and white?
Just for fun, I've accepted a challenge to white in a form of a mathematical expression the expression "Not all is black and white". At first, I thought it will be easy, but now I am stuck.
...
- 109
3
votes
0
answers
83
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What does this infinite series converge to? [closed]
$\sum_{n=1}^{\infty} \frac{(-2)^n(n!)}{n^n} = \space ?$
How would you figure this out? I only managed to show that it does converge and i wrote a python script that showed that it converges to roughly ...
- 475
1
vote
1
answer
46
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Attempt at proving uniform convergence of a sum
For a problem, I need to show
$$\lim_{b \rightarrow 1^-} \sum_{n=0}^{\infty}\frac{(-b)^n}{n+\gamma} = \sum_{n=0}^{\infty}\frac{(-1)^n}{n+\gamma}\hspace{10mm} \forall b \in [0,1]$$
My attempt at doing ...
- 609
1
vote
3
answers
78
views
Function equal to infinite series $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n+3)(2n+1)!}$
I'd like to know if there is a simple function equivalent of $$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n+3)(2n+1)!}$$
I recognize that it looks similar to $\frac{\sin{x}}{x}$, but with an extra $(2n+...
- 105
0
votes
0
answers
62
views
How many terms need to be added to approximate $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}$ with an error less to $10^{-5}$
How many terms need to be added to approximate $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}$ with an error less to $10^{-5}$
So since this is an alternating series we know that $$|R_n|=|S-S_n|≤a_{n+1}$$
...
- 87