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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0
votes
2answers
71 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?:
2
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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 ...
2
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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 ...
89
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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. ...
6
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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}...
31
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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 $\Psi_3(x)$ ...
-1
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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 ...
1
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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 ...
15
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0answers
224 views
+100

Limit associated with a recursion

If $z_n < 2y_n$ then $y_{n+1} = 4y_n - 2z_n$ $z_{n+1} = 2z_n + 3$ Else $y_{n+1} = 4y_n$ $z_{n+1} = 2 z_n - 1$ Consider the following limit: $$\lim_{n\rightarrow\infty} \frac{1}{n}\left(z_{n+1}...
4
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6answers
102 views

Convergence of complex series $\sum_{n=1}^{\infty}\frac{i^n}{n}$

Prove that the series $\displaystyle \sum_{n=1}^{\infty}\frac{i^n}{n}$ converges. Optional. find it's sum, if possible. Comments. I am aware of the general result about the convergence (not ...
2
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2answers
51 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(\...
3
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0answers
28 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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3answers
34 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 ...
-6
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0answers
32 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
votes
2answers
74 views

Limit of sum of areas of infinite amount of triangles

I apologize for the possible incorrect use of math terms since English is not my native language and I'm not a mathematician, but this issue came to my mind about a month ago and I was unable to solve ...
0
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1answer
65 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, $$...
1
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2answers
3k views

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 ...
3
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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 ...
9
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1answer
324 views

The Chudnovsky pi formula $1/\pi$ revisited

Define the constants, $$A=163\cdot1114806\\B=13591409\\C=640320$$ Given the binomial coefficient $\binom{n}{k}$, then we have the pi formulas, $$\frac{1}{\pi} =\frac{12}{(C)^{3/2}}\sum^\infty_{k=0} \...
0
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1answer
33 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.
3
votes
2answers
76 views

Why does $\frac{1+t}{(1-t)^3}=\sum_{n=0}^{\infty}(1+n)^2t^n$

Background: This is a step from a longer proof/exercise that $\sum_{n=1}^{\infty}\tau (n)^2/n^s=\zeta(s)^4/\zeta(2s)$ for $\sigma>1$ Expanding the sum and using counting I get: $\frac{1+t}{(1-t)...
3
votes
1answer
105 views

Determine the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$

I need to calculate the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$ . It seems very "similar" to Taylor expansion of functions arcsin(x) and its derivative for x = -2. It is known: $...
5
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1answer
34 views

Finding the geometric progression based on the given details

The sum of infinite number of terms of a GP is 4, and the sum of their cubes is 192. Find the series. The following image is solution from my book. My doubt is why is $r=-2$ rejected? Is there any ...
44
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4answers
16k views

Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $.

How does one prove the following limit? $$ \lim_{n \to \infty} \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$
1
vote
3answers
82 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 (...
7
votes
0answers
118 views

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 ...
0
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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 ...
1
vote
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'...
0
votes
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 ...
1
vote
0answers
93 views
+50

Prime number and Relationship of Sequences of period 4,5,and 6

Let $p$ be a prime number.($p \neq 2,3,5$) Let $t^+,t^-,a$ be sequences. $t^+_{k+5}=t^+_k,t^+_1=0,t^+_2=-1,t^+_3=-1,t^+_4=0,t^+_5=2$ $t^-_{k+5}=t^-_k,t^-_1=-1,t^-_2=0,t^-_3=0,t^-_4=-1,t^-_5=2$ $a_k=...
8
votes
2answers
93 views

Necessary condition of changing signs of a divergent series $\sum_{n=1}^{\infty}p_{n}$ to make it convergent,$p_{n}$ decreases and tends to $0$.

$p_{n}$ decreases and tends to $0$.$\sum_{n=1}^{\infty}p_{n}$ is divergent. We choose $\varepsilon_{n}=\pm 1$ to make $\sum_{n=1}^{\infty}\varepsilon_{n}p_{n}$ convergent.I want to prove that $$\...
0
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1answer
40 views

Compute the Taylor series for $f(x)= \frac{1}{2 - x}$

Where does this series converge? Knowing $\frac{1}{2 - x} = \frac{1}{2}\frac{1}{1 - x/2}.$ My taylor series is $\frac{1}{2}+\frac{1}{4}x+\frac{1}{8}x^2+\frac{1}{16}x^3+\frac{1}{32}x^4$ with center $...
-2
votes
2answers
72 views

What is the value of x in $2 = x + \sqrt{x + \sqrt{x + \dots}} $? [duplicate]

If $x + \sqrt{x + \sqrt{x + \cdots}} = 2$, what is the value of $x$?
-2
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2answers
43 views

TRUE/FALSE: If $\{a_n\}$ and $\{b_n\}$ are divergent sequences, then $\{a_nb_n\}$ is divergent. T or F? [on hold]

Is this statement "If $\{a_n\}$ and $\{b_n\}$ are divergent sequences, then $\{a_nb_n\}$ is divergent." True or False?
0
votes
2answers
34 views

How to deduce the sup of a function sequence $f_n$ without computing the derivative?

I came across the following function sequence: $$ f_n = e^{-nx}\sin(nx)$$ I'm asked if the following sequence converges uniformly on the set of positive real numbers $[0,\infty[$. I found 2 ...
16
votes
2answers
670 views

Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it ...
5
votes
2answers
556 views

I think I found a flaw in Riemann Zeta Function Regularization

I think I may have found a flaw in how Zeta Regularization works. As we all know, it's very famous for proving that $1+2+3+4+...=(-1/12).$ See here (5 rows of equations at the end of this post) •On ...
0
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2answers
43 views

When can we use the integral test?

I was going over my notes and I found that I wrote that we cannot use the integral test on the following series, why is that? $$ \sum \frac{5}{k^2 \ln(k)} $$ Isn't it both decreasing and positive? ...
140
votes
6answers
5k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: $$f(x)=\...
19
votes
1answer
424 views

Showing that $a_{n+1}=\frac{n}{a_n}-a_n-a_{n-1}$ with $a_0 = 0$ and $a_1=2\Gamma(\frac34)\big/\Gamma(\frac14)$ stays positive for $n\geq1$.

This posting consists of several mildly-related questions, motivated from this posting. The main object is the following sequence. $$a_0 = 0, \qquad a_1 = x, \qquad a_{n+1} = \frac{n}{a_n} - a_n - a_{...
21
votes
4answers
568 views

recurrence relation, all terms of the sequence positive

Let $a_1=a$, $a_2=\frac{1}{a}-a$, $a_{n+1}=\frac{n}{a_n}-a_n-a_{n-1}$ for $n=2,3,4,...$. Find all $a$ such that $(a_n)$ is a sequence of positive reals. My attempt was to look at $a_3=\frac{3a^2-1}{...
20
votes
2answers
469 views

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$?

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? In other words, does there exist an infinite sequence $(a_k)_{k \in \mathbb N_0}$ such that $$e^x = a_0 + \sum_{1 \leq k < \...
0
votes
1answer
51 views

Does the series $\sum_{n=1}^{\infty}\frac{\cos(n+x)}{n}$ converge uniformly?

Does the following series $$\sum_{n=1}^{\infty}\frac{\cos(n+x)}{n}$$ converge uniformly? I know the series converges pointwise since $\sum_{n}\frac{\cos n}{n}$ and $\sum_{n}\frac{\sin n}{n}$ converge....
-3
votes
0answers
36 views

Divergence of $\sum_{n=2}^\infty \frac{1}{\log n}$ [on hold]

I need to know the convergence or divergence $$\sum_{n=2}^\infty \frac{1}{\log n}$$ I want the solution using Comparison test or D'Alembert Ratio test.
2
votes
3answers
84 views

Is this a proof by induction question?

Happened to stumble across this question and to me it immediately made me assume it's a proof my induction question but doesn't seem to be so. Question : Show that for every natural $k$ and $n$ ...
2
votes
2answers
201 views

Sum of set bits in every element for a natural numbers

I was thinking of a mathematical puzzle with binary representation of numbers, but could not find a convincing answer myself. Here is the puzzle: Say for some number N, I want to find the sum of the ...
2
votes
0answers
52 views

Resources on operations on finine sequences [on hold]

More than $25$ years ago I came up with this problem. I know the solution empirically, but I want some help to understand it in a better way. Which theories deal with such a problem? Do you recommend ...
2
votes
1answer
40 views

Prove that $\sum_{n=1}^{\infty} \frac{z^2}{n^2 - z^2 }$ converges uniformly

I'm trying to show that the serie $\sum_{n=1}^{\infty} \frac{z^2}{n^2 - z^2 }$ converge uniformly over all compact subsets of $B(0,1)$. If we take a compact $K\subset B(0,1)$ for all $z\in K$ we have $...
6
votes
3answers
68 views

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 ...
0
votes
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 & ... &...