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

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28
votes
3answers
772 views

On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?

I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
28
votes
1answer
741 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
28
votes
2answers
1k views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
27
votes
4answers
2k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
27
votes
4answers
1k views

Evaluate:: $ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n+1}\left( 1 + \frac12 +\cdots + \frac 1n\right) $

How to evaluate the series: $$ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n+1}\left( 1 + \frac12 + \cdots + \frac 1n\right) $$ According to Mathematica, this converges to $ (\log 2)^2 $.
27
votes
2answers
4k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty ...
27
votes
1answer
3k views

Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$

Here is an exercise, on analysis which i am stuck. How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on ...
27
votes
2answers
7k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
27
votes
2answers
995 views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \{\mathrm{e}-(1+\frac1n)^{n}\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & ...
27
votes
1answer
2k views

Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent?

Is the series $$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$ convergent? If one want to use Abel's test, is $$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$ convergent? Thank you very much
27
votes
3answers
1k views

Asymptotic (divergent) series

MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in ...
27
votes
1answer
565 views

Show $(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdots)^2 = 1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49} + \cdots$

Last month I was calculating $\displaystyle \int_0^\infty \frac{1}{1+x^4}\, dx$ when I stumbled on the surprising identity: $$\sum_{n=0}^\infty (-1)^n\left(\frac{1}{4n+1} +\frac{1}{4n+3}\right) = ...
27
votes
2answers
1k views

Why do some Fibonacci numbers appear in an approximation for $e^{\pi\sqrt{163}}$?

It is rather well-known that, $e^{\pi\sqrt{43}} \approx 960^3 + 743.999\ldots$ $e^{\pi\sqrt{67}} \approx 5280^3 + 743.99999\ldots$ $e^{\pi\sqrt{163}} \approx 640320^3 + 743.999999999999\ldots$ Not ...
27
votes
1answer
667 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
26
votes
10answers
4k views

Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$

Prove that for all integers $n$, $n \geq 1$, $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$ How would I go about proving this?
26
votes
12answers
3k views

$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
26
votes
5answers
3k views

Find closed form for $1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$

Is there any closed form for the following? $$1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$$ I tried to find one, but I failed. I saw solution on Wolfram Alpha, but I didn't understand it: ...
26
votes
4answers
3k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
26
votes
4answers
896 views

How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$

Find this sum $$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$ My try: let $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$ then we have ...
26
votes
2answers
687 views

Computing $ \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$

I would like to compute: $$ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$$ I wanted to use Fubini's theorem for double series but $$ ...
26
votes
2answers
741 views

Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the ...
26
votes
2answers
479 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
26
votes
2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
26
votes
1answer
620 views

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
26
votes
2answers
346 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
26
votes
4answers
560 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...
26
votes
3answers
561 views

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following conjecture true? A conjecture : If $a_n$ is an integer, then $n\le 8$. I ...
26
votes
2answers
576 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, ...
26
votes
1answer
779 views

A question about series with a strange property.

Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges? This is the ...
26
votes
2answers
581 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
26
votes
0answers
941 views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
25
votes
6answers
1k views

How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?

I'd like to find out why \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align} I tried to rewrite it into a geometric series \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = ...
25
votes
4answers
520 views

Methods to find $\lim\limits_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $

What would you suggest here? $$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$
25
votes
2answers
424 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
25
votes
4answers
564 views

Does the series $\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$ converge?

$$\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$$ My guess is "yes", but I can't prove it.
25
votes
2answers
677 views

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? Does there exist a sequence with the same property but with each term ...
25
votes
1answer
713 views

Evaluate $\sum\limits_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$

How to find $$\sum_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$$ I try something like this: $$\begin{align*}\sum_{k=1}^{\infty} ...
25
votes
1answer
425 views

Does my “Prime Factor Look-and-Say” sequence always end?

I'm trying to create a challenge for PP&CG where the object will be to find the longest sequence in a given time, but I'm worried that there may be an infinite sequence that will ruin things. The ...
25
votes
1answer
488 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
25
votes
3answers
1k views

Integrating $\int_0^ex^{1/x}\ \mathrm dx$

Compute $$\int_0^ex^{1/x}\;\mathrm dx.$$ There is an analytical anti-derivative found in this answer. How does one compute this? Using the anti-derivative approach we have $$\int x^{1/x}\;\mathrm d ...
25
votes
0answers
1k views

Does this sequence have any mathematical significance?

Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you ...
24
votes
24answers
17k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
24
votes
6answers
1k views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
24
votes
6answers
1k views

Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$

Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
24
votes
3answers
1k views

Result of the product $0.9 \times 0.99 \times 0.999 \times …$

My question has two parts: How can I nicely define the infinite sequence $0.9,\ 0.99,\ 0.999,\ \dots$? One option would be the recursive definition below; is there a nicer way to do this? Maybe put ...
24
votes
1answer
455 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
24
votes
2answers
618 views

A double series yielding Riemann's $\zeta$

Can you give me some hints to prove equality: $$\sum_{m,n=1}^{\infty} \frac1{(m^2+n^2)^2} =\zeta (2)\ G-\zeta(4)=\frac{\pi^2}{6}\ G-\frac{\pi^4}{90}$$ where $\zeta (t):= \sum\limits_{n=1}^{+\infty} ...
24
votes
3answers
1k views

Apery's proof of the irrationality of $\zeta(3)$

I am trying to understand Apery's 1978 proof that $\zeta(3) = \displaystyle \sum_{n=1}^\infty \frac{1}{n^3}$ is irrational. The idea behind the proof is to find an 'accelerated' series for $\zeta(3)$ ...
24
votes
1answer
377 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
24
votes
2answers
529 views

Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?

I was flipping through Proofs Without Words (PWW) and saw many visual proofs for sequences and series. However, I saw none for $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ Are there any ...