# Tagged Questions

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### Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
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### On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
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### Prove $\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$ using Dominated or Monotone Convergence

Is there a way to prove that $$\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$$ via the Dominated or Monotone Convergence Theorem?
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### Prove the inequality based on an infinite series

Define $$f(x)=\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}.$$ It is easy to see that this series converges for $x\in(-1,1).$ Now we are asked to show that $(1-x)^2f(x)\geq x,$ for $x\in[0,1).$ I tried ...
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### Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
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### Sum of a convergent and divergent sequence [closed]

If $(x_n)$ is a convergent sequence and $(y_n)$ is a divergent sequence, show that their sum diverges.
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### Giving some hints [duplicate]

The sequence of $\begin{Bmatrix} {x}_{n} \end{Bmatrix}$ is strictly decreasing,$\quad \lim_{n\to\infty }{x}_{n}=0 \quad$,and$\quad \lim_{n\to\infty }{y}_{n}=0 .$ From the above ...
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### uniform convergence of a functional sequence

Is this sequence of functions $$f_n(x)=n^3x(1-x)^n$$ converges uniformly for $x\in[0,1]$. I need some help on this.
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### Can every positive real be written as the sum of a subsequence of dot dot dot

I answered this thing Infinite sum of prime reciprocals and now wonder what happens if we do not have such a strong condition as Bertrand's postulate. i have been fiddling with this, not sure either ...
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### Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
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### A sum for stirling numbers Pi, e.

In this identity $$1-e{}^{2} = \displaystyle \sum _{n=0}^{\infty } \frac{(-1)^n(\pi )^{2 n}} {(2 n)!}\sum _{k=0}^{2 n} (-1)^{k} S_2(2 n,1-k+2 n),$$ $S_2$ is a Stirling number of the second kind. ...
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### Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
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### the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
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### Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge? [closed]

I need to solve $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$ Does this converge or diverge and why?
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### How Find the sum $\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)$

Prove or disprove $$I=\sum_{n=2}^{\infty}\left(1+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots+\dfrac{1}{(2n-1)^2}\right) \binom{2n}{n}\dfrac{1}{2^{2n}(2n+1)}=\dfrac{\pi^3}{48}-\dfrac{1}{6}$$ My ...
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### Series of sequence converges?

Given the recursively defined sequence $$a_2 = 2(C+1)a_0$$ and $$a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}.$$ that I got from the Frobenius method applied to an ODE, ...
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### Proof regarding limits of functions and sequences

For a function f$:\mathbb{R} \rightarrow \mathbb{R}$ define what is meant by $\ f(x) \rightarrow \infty$ as $x \rightarrow -\infty$ and prove that it holds if and only if whenever ...
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### What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
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### What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
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### Prove that a sequence is decreasing

Suppose that 0 < a < 1 Show that {a^n} is a decreasing sequence. Yes, this is a homework question. I think I can solve it using induction, but I'm not sure.
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### Small question about strong convergence

I have a small question I have that $\lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
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### series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...