# 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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### Determine whether the series is converges or not.

$$\sum_{n=0}^\infty\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$$ The question is to determine whether the series converges or diverges. I’ve tried to use the integral test but I couldn’t figure it out.
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### The integral of an infinite sum

For the integral: $\int_{1}^{\infty}\sum_{n=x}^{\infty}\frac{1}{n^{4}}dx$ How does one go about calculating this? I've only learned how to do U-substitution, and I lack the knowledge to somehow ...
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### What does $\sum_{m=1}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$ converge to?

What does the following sum converge to as $n\rightarrow\infty$? $$\sum_{m=2}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$$ I think it converges to $-\theta^2/2$, but I am not ...
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### Show that the series converges and find the error

I have the following question here: a) Show that $\displaystyle \sum_{n=1}^{\infty} \left( \sin \left( \frac{1}{2n} \right) - \sin \left( \frac{1}{2n+1} \right) \right)$ converges b) Find an ...
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### Rewriting an explicit formula as an recursive formula

I have the following series: $x_n = \frac{n}{n+1}$ and I'm trying to rewrite it as a recursive formula. I wrote the first 5 elements: $1/2, 2/3, 3/4, 4/5, 5/6$ however I fail to find the recursive ...
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### Sum of finite series

The sum of series $\frac{8}{5} + \frac{16}{65} + ....+\frac{128}{2^{18}+1}$ is A) $\frac{540}{1088}$ B) $\frac{1088}{545}$ C) $\frac{1001}{500}$ D) $\frac{1013}{545}$ I am unable to figure out ...
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### How to show that this sequence really converges

How I'm supposed to show that the sequence $x_i$ , (i is an integer number) that is given below converges to $2$. $x_0=x_1=1$ and for $i\geq1$: $$x_{i+1}=x_i+x_{i-1}-(x_i \cdot x_{i-1})/2$$ By ...
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### Showing that this particular sequence is Cauchy using the definition

This is the following information that I've been given; $$a_{1}=1,\, a_{2}=4,\, a_{n+2}=\frac{2a_{n+1}+a_{n}}{3}$$ I need to show that $|a_{m}-a_{n}|<\epsilon \,,\forall n\geq n_{0}, m>n$ I ...
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### What is the value of $[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})]$? Where $[x]$ stands for the greatest integer function.

What is the value of $$[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})]?$$ where $[x]$ stands for the greatest integer function. P. S : I ran a code on my PC to find that the ...
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So I have the following question: Show that the following series are convergent: a) $\displaystyle\sum_{n=2}^{\infty} \frac{\log_n(n!)}{n^3}$ b) $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(3n-2)^{... 3answers 25 views ### How to determine$\text{max } |z^n + a|$for$|z| \leq 1$,$a \in \mathbb{C}$,$n \in \mathbb{N}_{\geq 1}$? Let$n \in \mathbb{N}_{\geq 1}$and$a \in \mathbb{C}$, determine$\text{max } |z^n + a|$for$|z| \leq 1$. I do not understand the question, what is it exactly that I need to determine? And how do I ... 3answers 42 views ### How to show if$\sum_{n=1}^\infty 7X_{n+1} + X_n$converges and$X_n \to 0$when$n \to \infty$, then$\sum_{n=1}^\infty X_n$converges? Note that$X_n$is just a sequence of real numbers. I'm a bit out of ideas because I'm used to having the extra hypothesis of$X_n$being of positive terms. Any hints or suggestions will be greatly ... 2answers 77 views ### Solving$T(n) = 2T(\frac{n}{4}) + \log n$Solve following recursive relation$T(n) = 2T(\frac{n}{4}) + \log n$without resorting to the master theorem. I've tried substitution method but it didn't work. I don't know whether there is a method ... 2answers 31 views ### Finding explicit formula of a series I'm trying to find the explicit formula of a series:$a_1=\frac{1}{2}, a_{n+1}=\frac{(n+1)^2}{n(n+2)}a_n$I calculated, that$,a_2=\frac{9}{6},a_3=\frac{3}{5},a_4=\frac{5}{8}$. However, I fail to ... 1answer 27 views ### Sequence convergent to$e$[duplicate] Suppose$\lim_{n\to\infty}na_n=0$. We want to show that $$\lim_{n\to\infty}\left(1+\frac{1}{n}+a_n\right)^n=e$$ I am trying to use squeeze them, using$-\frac{2}{n}<a_n<\frac{2}{n}$but could ... 0answers 29 views ### Finding the limit given$X(1)$and$X(n+1)$[duplicate] Given$X_1 = 0$, and $$X_{n+1} = \frac{X_n^2 + 1}{2}$$ how can I use this information to find $$\lim_{n\to\infty} X_n$$ I only really have a basic understanding of how limits work but I have no idea ... 0answers 22 views ### Prove a sequence monotone & seek limit Let$a_n=\sqrt[n]{n},b_n=\frac{a_{n+1}}{a_n}$. Prove that from$n=5$,$(b_n)_{n\in\mathbb N}^{\infty}$is increasing and find$\displaystyle\lim_{n\to\infty}b_n$This is a textbook problem in chapter ... 0answers 61 views ### Evaluating$\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx$How to evaluate $$\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx\ ?$$ I came across this integral while I was trying to calculate$\int_0^{1/2}\frac{\operatorname{Li}_2^2(x)}{x}\ dx$... 0answers 12 views ### Approximations for infinite sums of a sequence resembling an exponential power series Let$\alpha,\beta,\gamma>0$and consider a sequence$\{A_k\}_{k\in\Bbb N}$defined by$A_0=1$and $$A_{k}/A_{k-1}=\begin{cases}\frac{\alpha}{k\beta} & ,k\leq n\\\frac{\alpha}{n\beta+(k-n)\... 2answers 27 views ### Calculating the convergence order I have the iterative formula:$$x_{n+1}=x_{n-1}\cdot (x_n)^2$$How can i calculate the convergence's order when the series is not a constant but converge. I know that i have to substitute x_n=\... 0answers 38 views ### Find whether \sum_{n=2}^\infty\frac{1}{n\ln^2(n)} converges [duplicate] I tried to compare it with 1/n which I know diverges:$$\frac{1}{n\ln^2(n)} \le \frac{1}{n} \Leftrightarrow \\ n\ln^2(n) \ge n \Leftrightarrow \\ \ln^2(n) \ge1\Leftrightarrow \\ \ln(n) \ge 1 \... 1answer 41 views ### If$f(r)=r(r+1)(r+2)(r+3)$, simplify$f(r+1)-f(r)$and use your result to find If$f(r)=r(r+1)(r+2)(r+3)$, simplify$f(r+1)-f(r)$and use your result to find $$\sum_{r=1}^n r^3$$ I need help with the second part of the question because the book does not give any answers for the ... 4answers 43 views ### Geometric series weighted by a cosine function [Edited, corrected notation!] I'm dealing with following series $$f_d(\omega) := \frac{1-d}{1+d}\sum_{j=-\infty}^{\infty} d^j cos(2\pi j \omega) = \frac{1-d}{1+d}\Big[1 + 2 \sum_{j=1}^{\infty} d^j ... 1answer 46 views ### Limit of a series of 1 and -1. If I know that \lim_{n\to\infty}\sum_{i=1}^n y_i > 0, where y_i can take the value of -1 or 1, can I assume that the series \rightarrow \infty? In other words, if I know the limit is ... 1answer 34 views ### On the convergence of particular sequences I'm reading Knopp's book on infinite series. Chapter 2 which ends with the main results on convergent sequences (not series), suggests the following excercises (the author assumes known the simplest ... 2answers 38 views ### Find general term of recursive sequences x_{n+1}=\frac{1}{2-x_n}, x_1=1/2, Please help to solve: x_{n+1}=\frac{1}{2-x_n}, x_1=1/2, x_{n+1}= \frac{2}{3-x_n}, x_1=1/2 I know answers, but can't figure out the solution. The first one is obvious if you calculate first 3-5 ... 2answers 124 views ### On proving that \sum\limits_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24} Ramanujan found the following formula:$$\large \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$$I let e^{2\pi n}-1=\left(e^{\pi n}+1\right)\left(e^{\pi n}-1\right) to try partial ... 0answers 31 views ### Formula for a_n if a_n=(b(n-1)+c)a_{n-1}+(d(n-2)+e)a_{n-2} What is the general formula of a term a_n of a sequence of real numbers \{a_n\}_{n=0}^\infty satisfying$$a_n=(b(n-1)+c)a_{n-1}+(d(n-2)+e)a_{n-2}, \quad n\geq 2,$$where b,c,d and e are some ... 2answers 78 views ### On the conjecture that 1-\frac 13 + \frac 16 +\frac 1{10} -\frac 1{15}+\cdots = 1\frac 19 Conjecture:$$1-\frac 13 + \frac 16 +\frac 1{10} -\frac 1{15}+\cdots = 1\frac 19$$where the pattern of the signs is +,-,+,+,-,+,+,+,-,\cdots and the denominators are the triangular numbers. ... 1answer 114 views ### Can anyone help me find the next term for this sequence: 1, 2, 5, 8, 13, 17, 23, 30, 39? I generated this sequence while working on this problem. Say we have n people trying to pee at n urinals. The urinals and people are all at lattice points (people at (0,0),(1,0),(2,0),...,(n-1,0) and ... 3answers 99 views ### Find \lim\limits_{x\rightarrow 0^+}\frac{1}{\ln x}\sum\limits_{n=1}^{\infty}\frac{x}{(1+x)^n-(1-x)^n} Find$$\lim\limits_{x\rightarrow 0^+}\dfrac{1}{\ln x}\sum_{n=1}^{\infty}\dfrac{x}{(1+x)^n-(1-x)^n}.$$Consider$$f(x):=(1+x)^n,$$By Lagrange MVT, we can obtain$$\frac{2x}{f(x)-f(-x)}=\frac{1}{f'(\... 0answers 15 views ### Equating coefficient [closed] If$\sum\limits_{n = 1}^\infty {{P_n}} \left( t \right){z^n} = z\sum\limits_{n = - \infty }^\infty {{{\left( {\beta z} \right)}^n}} $,Equating the coefficient of${z^n}$on both sides for n=1,2,3…$\...
I want to know what is the sum of $\sum_{n=1}^{\infty} \frac{1}{n 4^{n}} x^{n}$? Thanks guys!