# 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.

43,950 questions
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### Determine this limit

how can I determine the following limit? $$\lim_{n\to\infty} \frac{\ln\left(\frac{3\pi}{4} + 2n\right)-\ln\left(\frac{\pi}{4}+2n\right)}{\ln(2n+2)-\ln(2n)}.$$ This question stems from this question. ...
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### Compact convergence of bivariate complex series

Let $b_n\in H(D(0,R))$ be a sequence of holomorphic functions on a disc of radius $R$ centered at $0$, $\Omega=D(0,R)\times D(0,S)$, $|w_0|<S$. Take a compactly convergent (that is, uniformly ...
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### Verify this sum

Apparently this sum has this closed form. $$\sum_{k=j}^{n}{k \choose j}{n \choose k}{n-j \choose k+j}={2j \choose j}{2(n-j)\choose n-j}{n-j \choose 2j}{n+j \choose j}^{-1}$$ How to show that this ...
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### Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

I have happened to have proved this sum while attempting to prove another summation. Let $$S_n=\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k$$ ${2k \choose k}$ is the ...
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### 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 $$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$$ where $\Psi_3(x)$ ...
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### necessary and sufficient conditions that a number being prime or prime of special form? [on hold]

I like to gather some statements about the properties of prime numbers or prime of the specific forms. For instance 1) A prime number is a whole number greater than 1 whose only factors are 1 and ...
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### Evaluation of the series

Given$$f(x) = \frac{2^x }{2^x +\sqrt{2}}$$ Then find $$S_n= \sum^{2n-1} _{r=1} 2f(\frac{r}{2n})$$ So I tried to evaluate it by adding and subtracting a $√2$ term from numerator , but it didn't help , ...
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### Is it possible to obtain the sum of this infinite series? [on hold]

Is it possible to obtain the sum of the infinite series: $$\sum_{n=1}^\infty \frac{c^n}{1-q^n}$$ where $0<c<1, 0<q<1$.
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### Couple of questions on Hurwitz theorem

Hurwitz theorem as stated in Hahn and Epstein's Classical Complex Analysis is as follows:
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### Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ doesn't converge, on the other hand it grows very slowly?...
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### Show $f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$.

If $f(x), g(x)$ are defined as following on $[0 , \infty)$, $$\tag 1 f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$$ $$\tag 2 g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}.$$ . Then how to ...
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### Continued fraction ${0,1,2,3,4,5,6,7,8,9,…}$ and Bessel function [duplicate]

I would like to ask how to get the following relation. 0 + K_{n = 1}^{\infty} \frac{1}{n} = 0 + \frac{1}{1+ \frac{1}{2 + \frac{1}{3+ \frac{1}{4+ \frac{1}{5 + ...}}}}} = \frac{I_1(...
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### An example of a product of two distinct convergent series that is divergent

Do you have an example of a product of two distinct convergent series $\sum x_n$ and $\sum y_n$ $(y_n ≥0)$ that is divergent?
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### $a_n$ is a positive integer for any $n\in \mathbb {N}$.

Let $(a_n)_{n\geq 1}$ be a sequence defined by $a_{n+1}=(2n^2+2n+1)a_n-(n^4+1 )a_{n-1}$. $a_1=1$, $a_2=3$. I have to show that $a_n$ is a positive integer for any $n\in \mathbb {N}, n\geq 1$. I ...
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### Geometric-like Sum over Primes

Is there a known way to evaluate sums of the form $$\sum_{p\text{ prime}} x^{p},$$ and are there any restrictions on the value of $x$ (e.g., $|x|<1$ for typical geometric series)? EDIT: The ...
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### Infinity in infinite series

We define infinite series as summation of terms in sequence. where sequence is defined to function from natural number to real numbers $f:\mathbb{N} \rightarrow \mathbb{R}$. But $\infty$ does not ...
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### Algorithm generating subset of primes, can we classify which of them or estimate how large percent of primes are generated?

Assume I have following algorithm: Two lists of numbers, first starting at 2, second starting empty. We now follow rule: Add a number to first list which makes difference with latest number the ...
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### Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
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### Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!}$ converge?

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!}$ converge? I have no idea how to do this. I have tried to use any trick I am aware of but can't figure this out. Can anyone help ...
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### How to find the limit sum of a series [on hold]

Suppose $$S_n = \lim_{n\to\infty}\frac{\exp(i/2)}{\sum_{j=1}^{i}\exp(j/2)} \ \ \ \text{where} \ \ i = 1,\ldots,n$$ Programatically, $S_n\approx 0.49$ but would I show this by hand?
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### Is the proportionality characteristic of this function being carried on?

$A=kx$ is a directly proportional function,where $A^2=B^2+C^2$.Does it necessarily mean $B$ and $C$ both vary directly with respect to $x$? If not, under what condition is this possible? Thank you in ...
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### How to express this function as power series? [on hold]

How to express this function as power series $\frac{x}{(2+x^2)^2}$
In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums $$I=-\frac{\... 2answers 33 views ### Determine the series whether convergence or divergence with using ratio rest. [on hold] This is the problem:$$\sum_{n=0}^\infty 3^n\sin((\frac{1}{4})^n)$$I can't prove the convergence of this series, how can we solve it? 3answers 41 views ### How to prove the double sum of combinations is 3^n I have a double sum of combinations as follow$$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$I guessed and tested that S = 3^n, but I have no idea how to prove this. Any help is ... 1answer 109 views ### What is the sequence of accumulation points in the 2-adic space, of the Collatz graph? In the orbit of the function 3x+2^{\nu_2(x)} through "accumulation points" of the Collatz graph I have: ?\mapsto\dfrac{-\langle2\rangle\cdot\{5,7\}}{9}\mapsto\dfrac{-\langle2\rangle}{3}\mapsto \... 2answers 47 views ### Evaluation of series \sum_{n=0}^\infty\frac{5n+1}{(2n+1)!} How to evaluate series$$\sum_{n=0}^\infty\frac{5n+1}{(2n+1)!}$$I tried to split the summation...but I failed. Please help 3answers 76 views ### Studying the character of \sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}} I have to study the character of this series$$\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}} with $\alpha$ a real parameter. Considering the Cauchy condensation test, the equivalent ...
Let´s assume I have a function $f(\theta)>0$ defined for $\theta<\pi$ and $\theta >0$. I want to find its Legendre polynomial decomposition \$f(\theta)= \sum_{l=0}^\infty f_l \, P_l(\cos{(\...