For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0
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2answers
34 views

Proof series decreases by induction

I have a question regarding a proof by induction. We have to see whether or not the following series converges. $$U_n = \frac{1 \cdot 4 \cdot 7 \cdots (3n - 2)}{2 \cdot 5 \cdot 8 \cdots (3n-1)}$$ I ...
0
votes
0answers
29 views

Proving absolute or conditionally convergence

I have the following partial series - $$ \sum_{n = 0}^\infty {a_n} $$ when $$a_n=\begin{cases} \frac{1}{n} &; \quad n \ \text{is even}\\ \frac{-1}{n^2} &; \quad n \ \text{is odd}\ . ...
4
votes
3answers
2k views

Notation for factorial-type pattern with a skip/step of two instead of one?

I came across a peculiar pattern when solving a recurrence relation today: Some sequence $a_n$ looks as such: $a_0 = 1$ $a_2 = \frac{1}{2 \cdot 1}$ $a_4 = \frac{1}{4 \cdot 2 \cdot 1}$ $a_6 = ...
-1
votes
1answer
19 views

Limit of a sequence an and bn [closed]

I have two sequences $a_n$ and $b_n$ and the limit of $a_n$ is $a \in \mathbb{R>0}$ and the limit of $b_n \to \infty $. Now I have to show that $a_nb_n \to \infty$. I can imagine that it this is ...
1
vote
0answers
45 views

Prove that $\sum\limits_{k=1}^{n}\frac{1}{2n+k}+\sum\limits_{k=1}^{n}\frac{1}{n+k}=\frac{3n}{3n+1}+\sum\limits_{k=1}^{n}\frac{2}{(3k)^3-3k}$

Ramanujan's identity (1) $$\sum_{k=1}^{n}\frac{1}{n+k}=\frac{n}{2n+1}+\sum_{k=1}^{n}\frac{1}{(2k)^3-2k}$$ I found this in the paper Genius of Ramanujan vs Modern Mathematical Technology by Robert ...
1
vote
2answers
40 views

Series converge/converge absolutely/diverge

I need to determine if the two series are converge/converge absolutely/diverge: $1.\sum^{\infty}_{n=2}\frac{\sin{n}}{n\sqrt{n}}+\frac{\cos{n}}{n\ln(n)} $. ...
0
votes
1answer
41 views

$\frac{\sin(nx)}{n^p}$ series convergence check

Check for which $p$ functional series $\sum {\sin(nx) \over n^{p}}$ converges. It is easy to see that for any p > 1 it does converge for any real x. Because of comparsion test. How to figure out other ...
4
votes
1answer
40 views

Show that if $(\sum x_n)$ converges absolutely and $(y_n)$ is bounded then $(\sum x_n y_n)$ converges

This is the exercise 2.7.6 of the book Understanding analysis of Abbott, I want a check of my proof and if is needed additional information to complete it. a) Show that if the sequence $(\sum ...
-3
votes
2answers
24 views

Determine whether the series is covergent or divergent [closed]

Test the covergence ofthe series $\sum_{0}^{\infty}\frac{5^n+5}{3^n+2} $
1
vote
1answer
34 views

Sequence existing for a set of conditions

Let function $f$ is continous and limited on the interval $(x_0, +\infty)$. Prove $\forall \ number \ T \ \exists \ sequence \ \{x_n\},\ \lim_{n \to\infty}{\{x_n\}} = +\infty $: ...
8
votes
2answers
233 views

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
0
votes
2answers
43 views

how many n-digit palindromes exist

I'd never seen this kind problem before, and don't know where to start. Any help is appreciated. Thank you very much! A palindrome is a number that is the same forwards and backwards. For example, ...
3
votes
1answer
52 views

How to prove that $E\subset [0,1]$ with some property is countable

Let $E$ be a subset of $[0,1]$. For every sequence $(a_n)$ who's elements are in $E$ and different from each other, the series $\sum\limits_{n=1}^{\infty} a_n$ converges. prove that $E$ is countable. ...
0
votes
0answers
45 views

Series $\sum \lambda^{n-k} c_k $ converges to zero

Let $(c_n)$ a sequence of real number, such that $\lim_{n \to \infty} c_n=0$, Let $0<\lambda<1 $ and $\lambda^nc_0+\lambda^{n-1}c_1+\cdots+\lambda c_{n-1}+c_n=y_n$ a sequence. I have to prove ...
1
vote
2answers
33 views

Limit of a sequence, possibly requires epsilon delta

Show that if $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ are sequences for which $\lim_{n\to\infty} a_n = 0$ and $\{b_n\}$ is bounded, then $\lim_{n\to\infty} a_nb_n=0.$ This is what I ...
13
votes
0answers
79 views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
1
vote
1answer
58 views

How to find the sum of the series $\sum_{n=1}^\infty \frac{2^n+8^n}{11^n}$?

$$\sum_{n=1}^\infty \frac{2^n+8^n}{11^n}$$ My textbook does not give any example to help solve problems like this, only geometric series and I do not believe this is a geometric series?
3
votes
1answer
34 views

Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$

I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$. The ...
8
votes
3answers
190 views

A Ramanujan-type identity: $11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$

Out of curiosity, why it is these sums yield a rational answer? $$11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$$ I found this identity ...
1
vote
2answers
41 views

Limit question on independent random variables (Exercise 4.2.4 from Grimmett and Stirzaker)

Let $\{X_r | r \geq 1\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) < 1$ for all $y$, and let $Y(y) = \min \{k | X_k > y\}$. Show that $$ ...
0
votes
0answers
44 views

Minimizing the “distance” between a finite set of elements in a finite length sequence.

Given a set of "options", $\{A,B,C,C\}$, I'd like to construct a certain kind of sequence of these elements. And example sequence would be: $ABCDABCD$ I define some average "distance" for this ...
-1
votes
1answer
55 views

Uniform convergence of series $\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $

I'm having trouble with uniform convergence. I need to prove that $$\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $$ converges locally uniformly in the half-plane $Re z >0$ and find its sum. ...
5
votes
4answers
244 views

Estimate the sum of alternating harmonic series between $7/12$ and $47/60$

How can I prove that: $$\frac{7}{12} < \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} < \frac{47}{60}$$ ? I don't even know how to start solving this...
6
votes
2answers
301 views

Is it bad to call series a generalization of sum?

In a recent question I asked why series has a name separate from that of sum, and the general answer was that a series does not have the nice properties of sum. Does this mean it is bad to call series ...
4
votes
4answers
94 views

How to prove that $\sum_{n=1}^{\infty} \frac{(log (n))^2}{n^2}$ converges?

$$\sum_{n=1}^{\infty} \frac{(log (n))^2}{n^2}$$ I know that this series converges (proof by Answer Sheet). However I need to prove it using comparison, integration, ratio or other tests. The ...
11
votes
3answers
1k views

Why is it called a series?

Why did we make a new name for infinite sum? Was something wrong with calling it an infinite sum, or is it highlighting a difference between finite and infinite?
0
votes
1answer
28 views

Bose function convergence

How can I show that this series is convergent for z=1 and z<1 and divergent for z>1 $$\sum _{P=1}^{\infty }\dfrac {z^{p}} {p^{3/2}}$$ Using the ratio test I've found: $$\lim _{p\rightarrow ...
1
vote
0answers
55 views

Comparison test and DCT

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
5
votes
2answers
94 views

Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $[0,1]$

Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $[0,1]$. I was thinking in the direction of taking the maximum value of each term $\frac{x}{1+n^2x^2}$, ...
3
votes
6answers
66 views

Trouble evaluating the $\lim_{n\rightarrow \infty}(\frac{n-1}{n+2})^{n+2}$

I am reviewing my calculus and am not sure why I got this limit incorrectly, I know via wolfram it should be $e^{-3}$. $$L=\lim_{n\rightarrow \infty}(\frac{n-1}{n+2})^{n+2}\Rightarrow ...
0
votes
0answers
24 views

Finite sum of $_{1}F_{2}$ hypergeometric functions

Could you help me with this finite sum? $$ \sum_{k=0}^{n}\binom{n}{k}\,_{1}F_{2}\left(\frac{n+1}{2},\frac{1}{2}+n-k,\frac{1}{2}+k,z\right), $$ where $_{1}F_{2}$ is a hypergeometric function? ...
0
votes
2answers
19 views

Sequence of partial sums of e in Q is a Cauchy sequence.

Verify that $X_n= \{ \sum_{i=0}^n$ $\frac{1}{i!}$} is a Cauchy sequence in $Q$ with the Euclidean metric. I can't figure out how to find an $N$ that makes this work. I figure that $d(x_n,x_m) < ...
1
vote
4answers
41 views

$\sum_{n\in\mathbb{N}}a_n$ conditionally convergent$\Rightarrow \nexists X\subseteq\mathbb{N}$ infinite s.t. $\sum_{n\in X}a_n$ absolutely convergent

I'm wondering about the following: if $\sum_{n\in\mathbb{N}}a_n$ is a conditionally (but not absolutely) convergent series then there does not exists $X$ infinite subset of $\mathbb{N}$ such that ...
1
vote
0answers
30 views

given geometric sequence an arithmetic sequence, find third sequence

I don't know where I got wrong with this problem. My answer is different from the answer sheet. To solve this problem, I used geometric sequence as $a, ar, ar^2 ..., ar^9$ and arithmetic sequence as ...
1
vote
1answer
29 views

How to check if this $\sum_{n=1}^{\infty}\lvert(-1)^{n+1}\sqrt{1-\cos(\frac{1}{n})}\rvert$ series converges absolutely?

The series is $\sum_{n=1}^{\infty}\lvert(-1)^{n+1}\sqrt{1-\cos(\frac{1}{n})}\rvert$ and I need to check if it converges absolutely. I know that it equals to ...
3
votes
1answer
51 views

Alternate Definition of Infinite Series Summation?

Question I was wondering if one could define the sum of conditional convergence without using the notion of before or after (time)? My Understanding We define the following partial sum: $$ S_n = ...
3
votes
0answers
70 views

Partitioning positive integers using digital rivers

I stumbled on a very simple computer science question from the British Informatics Olympiad for schools and colleges. Embedded in it is a very interesting numbers theory problem. Here is the ...
0
votes
1answer
43 views

Show that $\left\{\frac{1}{n^3-n+1}\right\}$ is a subsequence of $\{1/n\}$

Show that $\left\{\frac{1}{n^3-n+1}\right\}$ is a subsequence of $\{1/n\}$ Attempt: We know that if $\{a_n\}$ has a subsequence $\{{a_{n_k}}\}$ then the sequence of indices $\{n_k\}$ should be ...
0
votes
1answer
40 views

Series and countable sets

If $E$ is a sub-set of interval $[0,1]$, and also for every sequence ($A_n$) which all it's elements are different and are in $E$, the series $\sum A_n$ converges. How do you prove that $E$ is a ...
1
vote
1answer
26 views

What is the Laurent expansion of $f(z)=\frac1{z-3}$?

What is the Laurent expansion of $f(z)=\dfrac1{z-3}$? In the region, $|z-3|>0$ ? I just computed the Laurent expansion in the region $|z|>3$ by dividing the denominator by $\dfrac1z$ and ...
-3
votes
0answers
47 views

Evaulate the expression [closed]

$$x^2 + \frac53x^3 + \frac{23}{12}x^4 + \frac{119}{60}x^5 + \dots + 2\times \frac{5000!-1}{5000!}x^{5000}$$ How to evaluate the expression for $x=0.7893$ to the nearest $20$ decimal places?
0
votes
3answers
56 views

exponential function and mathematical induction

May I ask how to solve the problem? Use mathematical induction to prove that for $x\geq0$ and positive integer $n$, $$e^x \geq 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$$
2
votes
2answers
86 views

Decide whether the series ${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$ converges or diverges

Determine whether the series converges or diverges $${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$$ I was thinking I should use ratio test but I get an ugly sequence that I don't know how to ...
1
vote
2answers
34 views

Term-by-Term Differentiation and UNIFORM CONVERGENCE: True Relation

For a series with $\sum u_n'(x)$ not uniformly convergent, and If $f '(x) = \lim_{n\to\infty} f_n'(x) $ where $f(x)=\lim_{n\to\infty} f_n(x) $ and $ f_n(x) $ $=u_1+u_2+ . . . +u_n$ Then the ...
1
vote
2answers
42 views

How to prove $\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$ diverges?

$$\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$$ It seems clear to me that this series diverges because the dominant temr is $1/n^{2/3}$, a p-series with $p < 1$ However I need to prove ...
1
vote
2answers
36 views

How to prove $\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $ diverges?

$$\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $$ It seems clear to me that this seires diverges since the dominant term is $(3/2)^n$, a geometric series with $r > 1$ However I am required to ...
3
votes
2answers
229 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote the number of tosses until you see “$TH$” for the first time. For example, for the sequence $HTTTTHHTHT$, we needed ...
10
votes
2answers
150 views

Closed form for $\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{n\pi}-1}-16^k\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{4n\pi}-1}$

We took the idea from this Ramanujan's identity $$\frac{1^{13}}{e^{2\pi}-1}+\frac{2^{13}}{e^{4\pi}-1}+\frac{3^{13}}{e^{6\pi}-1}+\cdots=\frac{1}{24}$$ A few examples of Ramanujan-type identities ...
4
votes
1answer
31 views

Convergence of series with iterated $\ln$

Let us consider two function $\textrm{pln}_1\colon \mathbb{N}\to\mathbb{R}$ and $\textrm{pln}_2\colon \mathbb{N}\to\mathbb{R}$: $$\textrm{pln}_1(n) = n\cdot\ln n\cdot\ln\ln ...
1
vote
1answer
22 views

Show that any arithmetic progression contains a sequence of composites of arbitrary length

My question is inspired by this one: Arithmetic sequence whose any five consecutive elements contain a prime A more precise form: Let $(x_n)|_{n=1}^{\infty}$ be an arithmetic progression such that ...