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

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23
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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 ...
23
votes
1answer
1k views

Uniform convergence of $\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$

I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ : $$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$ Can someone help me with it? (I can't use Dirichlet' ...
23
votes
4answers
451 views

$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $

What would you suggest here? $$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$
23
votes
4answers
622 views

If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?

Let $a_n$ be a sequence of complex numbers that converge to zero. Can we always find $s_n \in \{-1,1\}$ such that $\sum_{n=1}^{\infty} s_n a_n$ converges? If the $a_n$ are real numbers, we can find ...
23
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 $\zeta(3)$ ...
23
votes
1answer
487 views

Prove that $\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $f:[0,1]\to\mathbb R$ given by $$ f(x)=\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ ...
23
votes
2answers
633 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 ...
23
votes
1answer
555 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 ...
22
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: ...
22
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} = ...
22
votes
3answers
929 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 ...
22
votes
4answers
1k views

Maybe a rather famous integral

How to evaluate : $$\int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x$$ Thx guys! I was wondering how would use a series expansion?
22
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: ...
22
votes
3answers
505 views

Compute $\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$

How could the series below be computed ? $$\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$$ It's not a series from a book, but a series I thought of many times, and I didn't manage to figure out what I should do ...
22
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
22
votes
1answer
483 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 ...
22
votes
1answer
499 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} ...
22
votes
1answer
744 views

Repeated Factorials and Repeated Square Rooting

I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
22
votes
2answers
526 views

Is $\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin x\cdots\right)\right)=\frac4{\pi}\sum\limits_{k=0}^\infty\frac{\sin(2k+1)x}{2k+1}$?

We can see intuitively that $$ f(x)=\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin{x}\cdots\right)\right)\right) $$ is the square wave with period $2\pi$ and has the ...
22
votes
2answers
1k views

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

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 ...
21
votes
17answers
3k views

How can I get sequence $4,4,2,4,4,2,4,4,2\ldots$ into equation?

How can I write an equation that expresses the nth term of the sequence: $$4, 4, 2, 4, 4, 2, 4, 4, 2, 4, 4, 2,\ldots$$
21
votes
21answers
12k 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? ...
21
votes
3answers
1k 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 ...
21
votes
4answers
2k 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 ...
21
votes
4answers
757 views

How to compute the series $\sum_{n=0}^\infty q^{n^2}$?

Let $q\in (0,1)$. Is there a way of computing the series $$ \sum_{n=0}^\infty q^{n^2} $$ explicitly? Is there at least a nice accurate estimate? All I could get is the estimate ...
21
votes
2answers
683 views

How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
21
votes
1answer
282 views

Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$?

Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$, where $\mu(n)$ is the Möbius $\mu$-function?
21
votes
3answers
1k views

Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) ...
21
votes
1answer
796 views

What is limit of $\sum \limits_{n=0}^{\infty}\frac{1}{(2n)!} $?

What is the limit of the series $1 \over (2n)!$ for n in $[0, \infty)$ ? $$ \sum_{n = 0}^{\infty}{1 \over (2n)!}$$ I've ground out the sum of the 1st 1000 terms to 1000 digits using Python, (see ...
21
votes
4answers
623 views

Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} ...
21
votes
2answers
762 views

Find the value of the sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

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 & V. ...
21
votes
2answers
544 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} ...
21
votes
2answers
545 views

an integer sum of products of tangents

This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that ...
21
votes
1answer
804 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
20
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?
20
votes
6answers
4k views

Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
20
votes
3answers
574 views

Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$

Does this weighted sum of reciprocals of Fermat numbers, $$ F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1} $$ have a nice closed form? Wolfram says it's $1$. Thanks.
20
votes
4answers
832 views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
20
votes
4answers
553 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
20
votes
2answers
328 views

Mystery about $\sum_{n\geqslant 1}2^{-n!}$

I was playing with some series when Wolfram told me that $$\sum_{n\geqslant 1}2^{-n!}=0.765625059604644775390625\color{Red}{000000000000}752316384526264\ldots$$ and my eyes obviously stopped at the ...
20
votes
3answers
980 views

Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$

For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate ...
20
votes
2answers
1k 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( ...
20
votes
2answers
367 views

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
20
votes
1answer
313 views

The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized

The evaluation, $$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$ was recently asked in a post by Chris here. I ...
20
votes
1answer
443 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]- ...
20
votes
1answer
602 views

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

Please determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$ converges. (Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ...
19
votes
12answers
711 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
19
votes
2answers
997 views

please solve a 2013 th derivative question?

$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $ Find $ f^{(2013)}(0) $ A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
19
votes
7answers
1k views

Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?

I am searching for the value of $$\sum_{n=k+1}^{\infty} \frac1{n^3} \stackrel{?}{=} \sum_{n = 1}^{\infty} \frac1{n^3} - \sum_{n=1}^{k} \frac1{n^3} = \zeta(3) - \sum_{n=1}^{k} \frac1{n^3}$$ For which ...
19
votes
2answers
349 views

How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number

I'm wondering if it's possible to write a theorem to prove or disprove the possibility of arranging a sequence of numbers (1,2,...n) such that the sum of any two numbers adds up to a prime number. An ...