All Questions
1,810
questions
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Sum sequence using Stolz–Cesàro
I have this sequence, and I need to find the convergence of the sum sequence.
The answer is - sum equal π/4.
But I tried to solve it by Stolz–Cesàro, as you can see in the picture, And what I got is ...
- 6,565
6
votes
4
answers
241
views
Evaluate $\int_{0}^{1}\{1/x\}^2\,dx$
Evaluate
$$\displaystyle{\int_{0}^{1}\{1/x\}^2\,dx}$$
Where {•} is fractional part
My work
$$\displaystyle{\int\limits_0^1 {{{\left\{ {\frac{1}{x}} \right\}}^2}dx} = \sum\limits_{n = 1}^\infty {\...
1
vote
1
answer
46
views
Simplifying $\frac{1}{2}\sum_{n=0}^{\infty}{(n+1)(n+2)(z^n+z^{n+1})}$
This is part of a larger problem where I am trying to prove $\sum_{n=0}^{\infty}{(n+1)^2z^n}=(z+1)/(1-z)^3$. Thus far I have used the derivatives of the geometric series to obtain $\frac{1}{2}\sum_{n=...
- 142
4
votes
2
answers
488
views
Sum $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$
I have the following infinite sum:
$$
\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}
$$
Because there is a $(-1)^n$ I deduce that it is a alternating series.
Therefore I use the alternating series ...
- 91k
8
votes
2
answers
244
views
How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$
Let us calculate the sum
$$
\displaystyle{\sum_{n=1}^{+\infty}\left(\frac{H_{n}}{n}\right)^2},
$$
where $\displaystyle{H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}}$ the $n$-th harmonic number.
My try
The ...
- 14k
2
votes
2
answers
172
views
how to turn the sum $\lim\limits_{n\to\infty}\left(\prod\limits_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ into integral with Riemann sum
I want to solve
$\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ with integration
I know that $\lim_{n \to \infty}\frac{b-a}{n} \sum_{k=0}^nf(a+\frac{b-a}{n})=\int_a^b f(x)dx$
so ...
- 6,565
4
votes
1
answer
125
views
Find value of this sum
Let $$\lim_{x\rightarrow 0}\frac{f^{}(x)}{x}=1$$
and for every $x,y \in \mathbb{R} $ we have:
$$f(x+y)=f(x)-f(y)+ xy(x+y)$$
Now Find :
$$\sum_{i=11}^{17}f^{\prime} (i)$$
I think this question is ...
- 2,813
2
votes
1
answer
177
views
Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $
Show that $$ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)}
$$
My attempt
Lemma-1
\begin{align*}
\frac{\sin(2nx)}{\sin^{2n}(x)}&=\...
- 40.1k
1
vote
2
answers
205
views
Which closed form expression for this series involving Catalan numbers : $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$
Obtain a closed-form for the series: $$\mathcal{S}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$$
From here https://en.wikipedia.org/wiki/List_of_m ... cal_series we know that for $\...
- 54.3k
0
votes
2
answers
200
views
how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$
Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$
My attempt
Let
$\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$
$$S=\sum\limits_{k=1}^{...
- 8,659
0
votes
0
answers
132
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
CommunityBot
- 1
4
votes
2
answers
200
views
How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
My attempt
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
$$= \sum_{n=1}^{\infty} \...
- 1,083
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 73.8k
32
votes
1
answer
818
views
On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$
Motivation
$\newcommand{Li}{\operatorname{Li}}$
It is already known that:
$$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$
$$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$
And by this question, ...
- 1,647
2
votes
2
answers
227
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Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
