All Questions
Tagged with real-analysis summation
1,083
questions
1
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2
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66
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Transforming two index sum into single index sum
I know this is a quite basic question, but I have been looking at it for more than I would like to admit and I simply don't see the solution. Namely, I have the sum:
$$\sum_{1\leq i < j \leq n} \...
- 172
3
votes
2
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123
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Does this sum converge? Does it converge absolutely? $\sum_{n=1}^{\infty} \frac{(-1)^n}{n-\sqrt{n-1}}$
Does this sum converge? Does it converge absolutely?
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n-\sqrt{n-1}}$$
I first checked absolute convergence. Taking the absolute value of the term, we get $\frac{1}{...
- 1,135
5
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1
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106
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Does this serie converge ? $\sum_{n=1}^{\infty} \frac{1}{n^3 - 5n}$
Does this serie converge ? $$\sum_{n=1}^{\infty} \frac{1}{n^3 - 5n}$$
We have $$n^3 - 5n < n^3 \implies \frac{1}{n^3-5n} > \frac{1}{n^3}$$
We know $\frac{1}{n^3}$ converges because it's a p ...
- 1,135
3
votes
2
answers
181
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Interchanging integration with summation on a specific example
Consider the infinite sum
$$S=\sum_{j=1}^\infty (-1)^{j +1}\frac{1}{2j+1} \int_0^1 \frac{1-x^{2j}}{1-x}dx$$
and the associated integral
$$L=\int_0^1 \sum_{j=1}^\infty\frac{ 1-x^{2j}}{1-x} (-1)^{j +1}\...
- 482
2
votes
1
answer
114
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Complete the proof of $ \int_{-\infty}^{\infty}{e^{-x^2/2}x^{2n} dx} = \sqrt{2\pi} \frac{(2n)!}{2^nn!} $
What I already know is that if $L(s) := \int_{-\infty}^{\infty}{e^{sx}e^{\frac{-x^2}{2}} dx}=\sqrt{2\pi}e^{\frac{s^2}{2}}$, then $L^{(2n)}(0)=\int_{-\infty}^{\infty}{e^{\frac{-x^2}{2}}x^{2n} dx}$. So ...
- 65
0
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Useful Partial Sums
The following formula: $$\sum_{k=m}^nf(k)=c(n-m)+\sum_{k=m}^\infty(f(k)-f(k+n))$$(Where $f\rightarrow c$) can be proven by telescoping the infinite sum in the RHS. The use of this formula is to expand ...
- 6,549
1
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3
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103
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Why $\sum_{k=1}^n \frac{a_k}{s-a_k}=s \sum_{k=1}^n \frac{1}{s-a_k}-n$ where is $s=\sum_{k=1}^n a_k$?
Why
$$\sum_{k=1}^n \frac{a_k}{s-a_k}=s \sum_{k=1}^n \frac{1}{s-a_k}-n$$ where is $s=\sum_{k=1}^n a_k$?
but not
$$\sum_{k=1}^n \frac{a_k}{s-a_k}=\sum_{k=1}^n \sum_{k=1}^n \frac{a_k}{s-a_k} \neq s \...
- 153
1
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1
answer
79
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Asymptotic equivalent of $\sum_{k=1}^n a^k k^{-1/2}$
I encountered recently the following partial sum $\sum_{k=1}^n a^k k^{-1/2}$ with $a$ a constant approximately equal to $2.955$.
I was wondering if there were any clever way to find an asymptotic to ...
- 762
1
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0
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67
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Proof of Cesàro summation
This is a proof I came up with while working on the textbook Understanding Analysis:
Supposing $x_{n} \rightarrow x$, we have that
$$s_n = \frac{1}{n} \sum_{k=1}^{n} x_k \rightarrow x$$
Let $\epsilon \...
- 402
1
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0
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50
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$\dfrac{\mathrm{d}^n}{\mathrm{d}x^n}\dfrac{e^{ax}}{\ln(cx)}$ and summation with Stirling number of the first kind
I would like to calculate the $n$-th derivative of $\dfrac{e^{ax}}{\ln(cx)}$
I tried to calculate it in this way:
$$(fg)^{(n)}(x)=\sum_{k=0}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$$
$$\frac{\mathrm{d}^...
4
votes
1
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89
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A conjecture involving series with zeta function
Recently, I tried to evaluate a limit proposed by MSE user Black Emperor. In the process of evaluating the limit, I have obtained the following equality.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 1,250
4
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0
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87
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How deduce $\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ from $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$?
I know that $\displaystyle\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ for any integer $n \geq 1$ is true.
Now, suppose that $n$ is odd, how show
$$
\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\...
- 323
4
votes
0
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135
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Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
0
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1
answer
56
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Simplify $ \sum\limits_{i=0}^l{n\choose i}{m\choose l-i}$
Is there a nice way to simplify $ \sum\limits_{i=0}^l{n\choose i}{m\choose l-i}$? I tried to tinker a bit with telescope sums but it did not get me nowhere...
- 4,564
1
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1
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74
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Estimating a sum with an integral to nearest power of $10$
Suppose I want to estimate the sum $\sum_{x = -1000}^{1000} \sum_{y: x^2 + y^2 < 10^6 } x^2$, or identically, $\sum_{y = -1000}^{1000} \sum_{x: x^2 + y^2 < 10^6 } x^2$, to the nearest power of $...
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