# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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### How to compute $\sum_{k=1}^{\infty}{(\zeta(2k)-1)}$

How to compute $\sum_{k=1}^{\infty}{(\zeta(2k)-1)}$, where $\zeta(s) :=\sum_{k=1}^{\infty} \frac{1}{n^s}$ with $s>1$. Here's my process, what am I doing wrong?:
4answers
2k views

### Find the $2011^{th}$ term of the sequence $2,3,5,6,7,8,10,11,…$

Find the $2011^{th}$ term of the sequence $2,3,5,6,7,8,10,11,...$ (a) $2056$ (b) $2011$ (c) $2013$ (d) $2060$ My Approach: Let $2,3,5,6,7,8$ be one set of numbers. In every 6 terms we reach 8 ...
3answers
68 views

### Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$

Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$ I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ...
6answers
8k views

### Can the product of infinitely many elements from $\mathbb Q$ be irrational?

I know there are infinite sums of rational values, which are irrational (for example the Basel Problem). But I was wondering, whether the product of infinitely many rational numbers can be irrational. ...
1answer
114 views

### Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

How to prove the following two sums \begin{align} \sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}&=2\operatorname{Li}_5\left(\frac12\right)+\ln2\operatorname{Li}_4\left(\frac12\right)-\frac{31}{32}...
4answers
993 views

### Closed-form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where $\Psi_3(x)$ ...
1answer
52 views

### The sequence 1/(n-1) is apparently unbounded yet convergent. What am I missing here?

For the sequence 1/(n-1): x1 is undefined. I think that makes the sequence unbounded. However I can prove from definition that the sequence is convergent. And there is the theorem that all convergent ...
2answers
67 views

### The series $\sum_{n=1}^{\infty} \frac {\cos n}{2n^{\alpha}}$ converges for $\alpha \in (0,1)$.

I'm trying to solve this problem: Let $\alpha>0$ and $a_{n}=\frac{\cos n}{2n^{\alpha}}$ for all $n\in\mathbb{N}$. Prove that the series $\sum_{n=1}^{\infty}a_{n}$ coneverges. For $\alpha>1$, we ...
0answers
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+100

2answers
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### An example of infinite resistor ladder, but with infinite resistance instead of finite

I know about a problem about resistance of infinite resistor ladder like this: The solution is the equality $$R_l=R+R+\frac{1}{\frac{1}{R}+\frac{1}{R_l}},$$ which is true as if we remove three ...
1answer
61 views

### How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$

How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$ Here's my progress: I suppose $\sum_{n,m=2}^{\infty}{n^{-m}}=\sum_{n=2}^{\infty}{\sum_{m=2}^{\infty}{\left(\frac{1}{n}\right)^m}}$, so we're looking at ...
1answer
324 views

0answers
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### Prove $\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}\ dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

Nice little generalization: $$\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}\ dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n},\quad a=0,1,2,...$$ The point of this post is to save us some ...
0answers
20 views

### How to estimate the rate of convergence of this sequence $c_{k}=\sum_{t=1}^{k}\alpha_{t}\beta^{k-t}$?

Let $c_{k}=\sum_{t=1}^{k}\alpha_{t}\beta^{k-t}$. Suppose that the sequence $\alpha_{t}$ is monotonically nonincreasing. $\sum_{t=1}^{\infty} \alpha_{t}^{2}<\infty$ $0< \beta <1$ Q1:How ...
5answers
62 views

### $\sum_{n=0}^{\infty} (2n+1)x^n$ Closed Form

$\sum_{n=0}^{\infty} (2n+1)x^n$ Closed Form I'm trying to find the closed form for the specified series. However, I'm having a bit of trouble doing so. I assume there's a technique here that I haven'...
5answers
56 views

### $\sum_{n=0}^{\infty} (n+1)^2 x^n$ Closed Form [duplicate]

$\sum_{n=0}^{\infty} (n+1)^2 x^n$ Closed Form I'm a bit stuck on finding the closed form here. I don't think I can use the technique of computing derivatives here directly. Could someone point me in ...
0answers
93 views
+50

3answers
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### Is it possible to utilize the convergence of the sequence $z_{n+1}=a/(1+z_n)$ to prove that the sequence $x_{n+2} = \sqrt{x_{n+1} x_n}$ is convergent?

I'm doing Problem II.4.6 in textbook Analysis I by Amann/Escher. For $x_0,x_1 \in \mathbb R^+$, the sequence $(x_n)_{n \in \mathbb N}$ defined recursively by $x_{n+2} = \sqrt{x_{n+1} x_n}$ is ...
1answer
25 views

### Vector Constructor Syntax Similar to Sigma

Is there a syntax like $$\sum_{i=0}^n i^2$$ but for constructing vectors instead of summing? Something like $$?_{i=0}^n i^2$$ would represent \begin{pmatrix}0 & 1 & 4 & 9 & ... &...