For questions about recurrence relations, convergence tests, and identifying sequences

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2
votes
6answers
329 views

Sequence sum question

I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$ Can anybody help me?
-4
votes
2answers
561 views

How many pairs of integers $(A, B)$ are there in the range $[1,\ldots, N]$, such that $\gcd(A,B) = B$?

I am given a positive integer $N$ ($N\leq 10^9$). How many pairs of integers $(A, B)$ exist in the range $[1,\ldots, N]$ such that $\gcd(A,B) = B$?
39
votes
3answers
4k views

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
43
votes
4answers
2k views

How to sum this series for $\pi/2$ directly?

The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} ...
41
votes
4answers
3k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
24
votes
5answers
691 views

About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$

How to prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$ $H_n$ is the n th harmonic number
42
votes
2answers
1k views

Is there a function with the property $f(n)=f^{(n)}(0)$?

Is there a not identically zero, real-analytic function $f:\mathbb{R}\rightarrow\mathbb{R}$, which satisfies $$f(n)=f^{(n)}(0),\quad n\in\mathbb{N} \text{ or } \mathbb N^+?$$ What I got so far: Set ...
26
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
23
votes
5answers
1k views

How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

I would like to investigate the convergence of $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$ Or more precisely, let $$\begin{align} a_1 & = \sqrt 1\\ a_2 & = \sqrt{1+\sqrt2}\\ a_3 ...
12
votes
2answers
616 views

Sum of the squares of the reciprocals of the fixed points of the tangent function

The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...
19
votes
4answers
484 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
13
votes
3answers
298 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
11
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
10
votes
2answers
453 views

How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$

How can I evaluate the following series: $$1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}+\frac{3\cdot 5\cdot 7\cdot 9}{4\cdot 8\cdot 12\cdot 16}+\cdots$$ In one ...
18
votes
4answers
1k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
11
votes
9answers
2k views

Motivating infinite series

What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, ...
11
votes
2answers
341 views

How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?

It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
7
votes
2answers
389 views

Proving a function is continuous on all irrational numbers

Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb ...
7
votes
2answers
699 views

Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$

We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$
13
votes
2answers
1k views

Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
13
votes
6answers
566 views

How closely can we estimate $\sum_{i=0}^n \sqrt{i}$

By looking at an integral and bounding the error?
10
votes
1answer
424 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converges? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
5
votes
3answers
324 views

sum of arctangent

Here is an interesting topic. It comes from evaluating $$\sum_{k=1}^{\infty}tan^{-1}\left(\frac{1}{k^{2}}\right)$$ I managed to dig up an old paper I have on the sum of arctans by Boros and Moll. ...
4
votes
2answers
247 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
12
votes
4answers
644 views

How to prove by arithmetical means that $\sum\limits_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum\limits_{k=1}^{\infty}\frac{1}{k^{2}}$

I've been trying to prove, by arithmetical means, that $$\sum_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum_{k=1}^{\infty}\frac{1}{k^{2}}$$ without success. When I say "by arithmetical ...
9
votes
2answers
467 views

Sum of the reciprocal of sine squared

I encountered an interesting identity when doing physics homework, that is, $$ \sum_{n=1}^{N-1} \frac{1}{\sin^2 \dfrac{\pi n}{N} } = \frac{N^2-1}{3}. $$ How is this identity derived? Are there any ...
9
votes
2answers
2k views

Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?

Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges. I calculated $$ ...
8
votes
4answers
311 views

Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$

This is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea ...
7
votes
1answer
566 views

Another way to go about proving the limit of Fibonacci's sequence quotient.

It is not difficult to inductively prove that $$\eqalign{ & \phi = \phi + 0 \cr & {\phi ^2} = \phi + 1 \cr & {\phi ^3} = 2\phi + 1 \cr & {\phi ^4} = 3\phi + 2 ...
5
votes
3answers
272 views

Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?

Discuss the convergence of the sum: $$\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}.$$ My answer so far: It does not converge absolutely since $$\left | \frac{(-1)^k}{\sqrt{k}+(-1)^k} \right ...
9
votes
1answer
178 views

Divergent series of positive numbers

So I've been trying to figure out how to prove the following. Let $(a_n)$ be a sequence of positive numbers such that $\sum_{n=1}^\infty a_n =\infty$, and define $s_n=\sum_{i=1}^n a_n$. Then ...
9
votes
2answers
1k views

Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $[0,2\pi]$

This question arises because of a problem I was doing (Bartle 3rd edition, section 9.4 problem 3). It was like this. Given $a_n$ a decreasing sequence of positive numbers and suppose that ...
8
votes
4answers
407 views

Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$

For $x>0$ and $y>x+1$, how do we prove that $$\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)} = \frac{x}{y-x-1}$$
6
votes
1answer
1k views

Levin's u-transformation

Suppose I'm given a very slowly converging sequence $\sum_k a_k$. In the literature, the Levin u-transformation is mentioned as a good universal technique for convergence acceleration. I have ...
5
votes
4answers
139 views

Proof of $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$

Why would $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$ be true where $(a_n)$ is a sequence in $\mathbb{R}$? Edit: Let all $a_n$ be positive.
5
votes
5answers
1k views

Difficulties in a proof by mathematical induction (involves evaluating $\sum r3^r$).

Please help. I've been stuck on this for 2 days. Haven't found any easy explaining text. The question is : Prove by mathematical induction that : $$ \sum_{r=1}^n r3^r = \frac{3}{4} \left[ 3^n \left( ...
5
votes
3answers
434 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
2
votes
2answers
896 views

$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$

$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within ...
14
votes
5answers
1k views

$S=1+10+100+100+10000+… = -1/9$? How is that

its written that if you multiply the sum above by -9 and use the distributive law, all terms except "1" will cancel. I can't see that. I know that this is a divergent series. (the article I was ...
10
votes
2answers
462 views

For $a_n>0$ such that $\sum a_n $ converges, show that there exist $c_n>0$ such that $c_n\to \infty$ and $\sum a_n c_n$ is finite.

A problem From PhD Pre lims Exam: Let $ a_{n} > 0 $ for all $ n\in\mathbb{N}$ such that $\sum a_{n} $ converges. Show that there exist $ c_{n} > 0 $ ($n\in\mathbb{N}$) such that $ \lim ...
8
votes
1answer
200 views

Function mapping challange

For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$ and $f(k) = f(f(k + 1))$. This is ...
7
votes
3answers
294 views

Value of $\sum_{k=1}^{\infty}\frac{1}{k^2+a^2}$

So my question is to find the value of $\sum_{k=0}^\infty\frac{1}{k^2+1}$ and more generally $\sum_{k=0}^\infty\frac{1}{Q(k)}$ where Q is a quadratic polynomial with no zeroes on the integers. I can ...
6
votes
2answers
166 views

Checking my understanding: $1 - 1 + 1 - 1 + 1 - … = \frac{1}{2}$

I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows: Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...
5
votes
1answer
237 views

Convergence of series involving in iterated logarithms $\sum \frac{1}{n(\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k} }$

What is the quickest way to show when $$ S(\alpha_1,\alpha_2,\cdots,\alpha_k) = \sum\limits_{n=3}^\infty \frac{1}{n (\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k}} $$ converges, where $\log^k(n)$ ...
5
votes
2answers
861 views

Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$

How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in ...
5
votes
3answers
5k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
4
votes
2answers
260 views

General question on relation between infinite series and complex numbers

This is a strictly preliminary question. I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site. I'm trying to understand how the study of the ...
3
votes
1answer
193 views

Computing sums of divisors in $O(\sqrt n)$ time?

I have a sequence: $1,3,5,8,10,14,16,20,23,27,\dots$ I know that the recursive relation is: $$p[i] := p[i-1] + \text{number of factors of $i$}, \quad \text{with $p[1]=1$.}$$ How do I solve this ...
3
votes
4answers
511 views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
2
votes
1answer
112 views

Elementary proof that $a_n \to a \implies a_n^r \to a^r$ for $r \in \mathbb{Q}$

I've been looking for an elementary proof of the fact that rational exponentiation is continuous (for sequences). Let $r = p/q \in \mathbb{Q},$ with $p,q \in \mathbb{N}.$ Assuming $|a_n - a| \to 0,$ ...