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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21 views

Let [$c_n$]{for $n$ from $0$ to $+\infty$} be a sequence of real numbers.

It is known, that $c_n>1$ for any $n$ and that $\prod_{i=0}^\infty c_i$ diverges to $\infty$ Is it true that $\sum_{i=0}^\infty ((1/c_i)-(1/(c_ic_{i+1}))$ also diverges to infinity?
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1answer
39 views

Convergence for some series

I have to examine if the following series are convergent: (1) $ \sum_{n=1}^{\infty} (\sqrt {n^2+1}-n)$ we have that $$\sqrt{n^2+1}-n=\sqrt{n^2+1}-n\frac{\sqrt{n^2+1}+n}{\sqrt{n^2+1}+n}=\frac1{\sqrt{...
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0answers
34 views

Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

Notation. This question deals with series of real terms $\{a_n\}_{n\in\Bbb N}$ such that $\limsup_{n\to\infty}\sqrt[n]{|a_n|}=1$: the following terminology is used. If $\lim_{N\to\infty}\sum_0^N a_n=\...
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0answers
29 views

Sum of products of incomplete harmonic series

Let $n,k$ be constant integers. What is as tight upper bound on $S$: $$S_{n,k}=\sum_{i_1=1}^{n-k}\sum_{i_2=1}^{n-i_1}\dots\sum_{i_k=1}^{n-i_{k-1}} \frac{1}{i_1 i_2\dots i_k}$$ It is easy to show that ...
2
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1answer
68 views

Slowest diverging sum of reciprocals of integer sequence [duplicate]

How can you construct a series $$\sum_{n=1}^\infty \frac 1{a_n}$$ which diverges very slowly to infinity where $a_n$ is always a positive integer and $a_{n+1}>a_n$? I know that if $a_n=p_n$, where $...
3
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2answers
56 views

Coefficients of polynomial $f_n(q)=\prod_{k=1}^{n}(1+q^k)$

What is the general formula for the coefficient $c_n(k)$ where $$f_n(q)=\prod_{j=1}^{n}(1+q^j)=\sum_{k=0}^{n(n+1)/2}c_n(k)q^k?$$ I came across this problem while researching $q$-analogs. Indeed, ...
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2answers
85 views

Is $-\int_{0}^{\infty} \bigg( \exp(-\dfrac{\pi}{4}x²)) \bigg) \;dx+\bigg(\sum_{n=0}^{\infty }\bigg( \exp(-\dfrac{\pi}{4}n²)\bigg)=\dfrac12$ true?

The following sum may it is easy for computation $$-\int_{0}^{\infty} \bigg( \exp(-\dfrac{\pi}{4}x²)) \bigg) \;dx+\bigg(\sum_{n=0}^{\infty }\bigg( \exp(-\dfrac{\pi}{4}n²)\bigg)=\dfrac12$$ The sum ...
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1answer
24 views

Techniques for Testing Convergence of Certain Sums

Occasionally I've encountered sums of the form $$ \lim_{n\to \infty} \sum_{k = 0}^n f(n,k) $$ Are there known techniques for testing the convergence of such sums? For example, for sums of the form $\...
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1answer
48 views

Equivalence of Two Forms of the Exponential Function [duplicate]

One encounters the following definitions for $e^x$ \begin{align} e^x &= \sum_{n = 0}^\infty \frac{x^n}{n!} \\ e^x &= \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^n \end{align} One can ...
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0answers
25 views

Limit with sequences [on hold]

Evaluate $\lim_{\omega \rightarrow \infty } \sum_{j=1}^{\omega} (\omega^{-\omega^{j}})\cdot \prod_{j=1} j^{j} $
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3answers
36 views

Doubt regarding definition of converging sequence.

I am currently studying about converging sequence in my Real Analysis class. The definition of a converging sequence is A sequence of real numbers converges to a real number a if, for every ...
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2answers
43 views

limit of sequence defined by rationals and fractional parts

Let $a_0$ be a positive rational number. And for natural numbers $n$, define a sequence as $a_n=a_{n-1}/(1-\{a_{n-1}\})$ where $\{x\} = x-[x]$ , the fractional part of $x$. Then I have to show that ...
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2answers
184 views

Non standard solution to $f(x) = \frac{1}{2}\Big(f(\frac{x}{2}) + f(\frac{1+x}{2})\Big)$

This functional equation appears in the following context. Let $\alpha\in[0,1]$ be an irrational number (called seed) and consider the sequence $x_n=\{2^n \alpha\}$. Here the brackets represent the ...
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3answers
72 views

Does there exist a sequence that has countably infinite convergent subsequences?

I know that for every natural number $n$ there is a sequence with exactly $n$ convergent subsequences, where I consider two subsequences to be the same if they are equal as sequences (even if they ...
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1answer
45 views

Why does the concept of limits necessitate that infinite series can be assigned a value?

When summing an infinite series such as $1/2+1/4+1/8+1/16+...$ I have often seen the following sort of argument: The partial sums of the series cannot equal more than 1, as when you add another term ...
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3answers
48 views

Find an equivalent sequence

Consider $ u_n = (n+1)^{1/n+1} - n^{1/n} $ Find an equivalent sequence at infinity. (meaning $ u_n / y_n \rightarrow 1 ) $ I tried doing : $ u_n = e^{ \frac{ln(n+1)}{n+1}}(1 - e^{\frac{ln(n)}{n} -...
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0answers
35 views

Determining the convergence/divergence of recursive sequence

Is it always possible to define a sequence using both $n^{th}$ term formula and recursion formula? For example: $a_1=3$, $a_{n+1}=a_n+3$ defines the sequence $\{3,6,9,12,...\}=\{3n\}$ I am asking ...
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2answers
31 views

Formula for sequence of numbers

I am trying to find a somewhat nice general formula for the following sequence. $2,2,6,6,10,10,14,14,\dots$ I found one on wolfram alpha, however I am not sure how to derive it; I am also wondering ...
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2answers
52 views

For $x_0 \ge 1$, the sequence $(x_n)$ defined recursively by $x_{n+1} = (x_n +1/x_n)/2$ converges to $1$

I'm doing Problem II.3.4 in textbook Analysis I by Amann/Escher. After elementary transformations, the problem is equivalent to the below theorem: Theorem: For $x_0 \ge 1$, the sequence $(x_n)$...
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1answer
33 views

Limit comparison test for series

I'm confused by one thing lct states that if you have two series, $a_n$ and $b_n$ if taking the limit of two (where $bn$ is the decent comparison where it's positive for all n meaning it's limit ...
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2answers
34 views

Sequence converges means it is bounded above and below

What happens in a situation where $\{a_n\}$ has an asymptote, will there ever be a case? Because if there is one then it's no longer bounded.
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2answers
39 views

Convergent subsequence of $\sin(n^2)$

What can we say about the convergent subsequences of $\sin(n^2)$ whose existence is guaranteed by the Bolzano-Weierstrass theorem? Can we, as a corollary, claim that for all $\epsilon \gt 0$ there ...
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0answers
33 views

sum function of power series $\sum_{n=0}^{\infty}\frac{(2n)!!}{(2n+1)!!}x^{2n+1}$ [on hold]

I have a power series as shown below $$\sum_{n=0}^{\infty}\frac{(2n)!!}{(2n+1)!!}x^{2n+1}$$ How to get its sum function? Any help will be appreciated.
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1answer
47 views

How to derive this partial sum about a harmonic series?

I've recently run across an equation in a proof that I'd love to get help with. It argues that $$\sum_{i=1}^{N} i \frac{1}{iH_{N}} = \frac{N}{H_{N}}$$ Where $H_{N}$ is the Nth Harmonic Number (sum ...
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1answer
64 views

Check the convergence of $\sum_{n=1}^{\infty} \int_{0}^{1/n} \sqrt {x}/ (1+x^{2})dx$

Check the convergence of $$\sum_{n=1}^{\infty} \int_{0}^{1/n}\frac{\sqrt {x}}{1+x^{2}}\;dx$$ I think to solve this problem, first, I need to solve this integral. Then I will have this series in terms ...
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5answers
62 views

prove that sequence $\frac{n^2+1}{n+4}$ diverges

Can somebody help me prove this sequence? I've tried using Comparison Theorem and ended up with $a_n = \frac{n^2+1}{n+4} < \frac{n^2}n = n > N = M$. So I choose $N = M$. But I don't think ...
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3answers
43 views

Visual Interpretation for the Sum of a Finite Geometric Series

I'm interested in intuitive visual explanations for the sum of a finite geometric series. I know there are some pretty "intuitive" explanations out there (including some on this site), but I haven't ...
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0answers
91 views

Finding sum of $\sum\limits_{r=1}^{n}{\frac{r+1}{{{r}^{2}}+1}}$ [on hold]

$$\sum\limits_{r=1}^{n}{\frac{r+1}{{{r}^{2}}+1}}$$. since this sum is not till infinity, then can we solve for this?
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1answer
53 views

Harmonic numbers converging to any number

$x_n$ is equal to $x_{n-1}-\frac 1n$ if $x_{n-1}+\frac 1n \ge a$ and is else equal to $x_{n-1}+\frac 1n$ where $a$ is any real number and $x_0$ is a rational number (its usually defined by $\lfloor a \...
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2answers
66 views

How to flip all fractions in the power series for $\ln(1 + x)$?

I am trying to evaluate this using power series: $$1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} + \dots$$ By using the power series for $\ln(1 + x)$, I have recognized that dividing through by $x$ ...
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1answer
67 views

If the series $\sum_1^\infty a_n$ converges, then so does $\sum_1^\infty \frac{{a_n}}{n} $

The problem: Show that if $\sum_{1}^\infty a_n$ converges and $a_n ≥ 0$ for all $n ≥ 1$, then $\sum_1^\infty \frac{a_n}{n}$ also converges. Is the statement true without the hypothesis $a_n ≥ 0$ ? ...
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0answers
27 views

Finding limits will sequences [duplicate]

Let ${{a}_{1}},{{a}_{2}},......{{a}_{n}}$ be sequence of real numbers with ${{a}_{n+1}}={{a}_{n}}+\sqrt{1+{{a}_{n}}^{2}}$ and ${{a}_{0}}=0$ . Prove that $\underset{n\to \infty }{\mathop{\lim }}\,\...
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2answers
78 views

Find the sum $\sum_{n=1}^{\infty}\frac{ 1}{n^{2}(2n-1)}$ [on hold]

Find the sum $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2}(2n-1)}$ $= 1 + \dfrac{1}{3\cdot 2^{2}} + \dfrac{1}{5\cdot 3^{2}} + \dfrac{1}{7\cdot 4^{2}} \ldots$ Then, I'm stuck. I didn't find any ...
0
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1answer
38 views

If $\sum a_{n}$ is conditionally convergent. For $p > 1$, $\sum n^{p} a_{n}$ is divergent. (true/false)

If $\sum a_{n}$ is conditionally convergent. For $p > 1$, $~~\sum n^{p}~ a_{n}~~$ is divergent. Is the above statement is true or false $~?$ I think the statement is true. Since, I couldn't ...
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1answer
64 views

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

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 ...
4
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4answers
116 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 ...
1
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2answers
90 views

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?:
3
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1answer
47 views

How to find the value of Grandi's series using Ramanujan's summation

I can't figure out how to solve the infinite sum of $\sum ^{\infty }_{n=0}\left[( -1)^{n}\right]$ I know that Srinivasa Ramanujan solved it and I couldn't figure it out with Ramanujan's summation. ...
1
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2answers
79 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 ...
6
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1answer
140 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}...
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0answers
37 views

True/False: Suppose that $\{a_n\} \subset \mathbb{R}$. If $a_n$ is bounded for all $n \in \mathbb{N}$ and $\lim a_n =0$ then $\{a_n\}$ converges [on hold]

I know that $\lim a_n = 0$ is the Vanishing Condition, but I cannot figure out if it implies that sequence (edit: series) $a_n$ converges or the other way around. Any thoughts?
2
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2answers
57 views

Does $(f_n)=(n\sin(\frac{x}{n})-x)$ converge uniformly on $[-a,a]$ for $a\geq0$?

I'm trying to solve the next problem: Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a sequence of functions such that $f_{n}\colon\mathbb{R}\to\mathbb{R}$ is given by $f_{n}\left(x\right)=n\sin\left(\...
1
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3answers
37 views

Construct $b_n$ so that $\sum_{n=0}^{\infty}b_n=B$

There are given two real numbers, convergent series $\sum_{n=0}^{\infty}a_n=A$ and $\sum_{n=0}^{\infty}c_n=C$, such that $a_n<c_n\:\forall n\in\mathbb{N}$. Let $B\in(A,C)$; Construct $b_n$ such ...
0
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1answer
68 views

Uniform convergence of $\sum_{n=1}^{\infty} \dfrac{z^{n}}{1+z^{2n}}$ [duplicate]

Show that the series $\sum_{n=1}^{\infty} \dfrac{z^{n}}{1+z^{2n}}$ converges uniformly over the compact subsets of $\{z\in \mathbb C : |z|\neq 1\}$. if we assume $|z| = 1$. Then, by the limit test, $$...
3
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1answer
62 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 ...
3
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1answer
42 views

Conditions for Differentiable Limit Theorem

I know that if $(f_n)$ is a sequence of differentiable functions on $(a,b)$ with pointwise limit $f$ and if $f'_n \rightarrow g$ uniformly the $f$ is differentiable on $(a,b)$ and $f' = g$. I want to ...
0
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1answer
37 views

A sequence $x_n$ has a cauchy subsequence ii it has a subsequence satisfying the following property [on hold]

I am trying to solve this problem but could not make an idea. Please give some hint for the problem.
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3answers
87 views

Find the sum of the series $1^3 + 3\cdot 2^2 + 3^3 + 3\cdot 4^2 + 5^3 + 3\cdot 6^2…$ up to $n$ terms

Find the sum of first $n$ terms of the series $1^3 + 3\cdot 2^2 + 3^3 + 3\cdot 4^2 + 5^3 + 3\cdot 6^2...$ When $n$ is even. When $n$ is odd. This sum can be written as $$\sum_{1}^n (...
0
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5answers
57 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 ...
1
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5answers
63 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'...