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

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310
votes
10answers
20k views

My son's Sum of Some is beautiful! But what is the proof or explanation?

My youngest son is in $6$th grade. He likes to play with numbers. Today he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, ...
246
votes
23answers
22k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
79
votes
4answers
2k 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: ...
77
votes
13answers
6k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
75
votes
8answers
4k views

Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$

To prove the convergence of $$\sum_{n=1}^{\infty} \frac1{n^p}$$ for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test. I am wondering if there is a ...
73
votes
6answers
7k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
70
votes
6answers
4k views

Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
70
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
69
votes
6answers
2k views

Contest problem about convergent series

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a set of positive real numbers and the series $$\sum_{n = 1}^\infty ...
68
votes
17answers
4k views

Proving the identity $\sum\limits_{k=1}^n {k^3} = {\Large(}\sum\limits_{k=1}^n k{\Large)}^2$ without induction

I recently proved that $$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2 $$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
64
votes
8answers
3k views

Evaluating $\lim_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. I ...
63
votes
13answers
11k views

How to find a general sum formula for the series: 5+55+555+5555+…?

I have a question about finding the sum formula of n-th terms. Here's the series: $5+55+555+5555$+...... What is the general formula to find the sum of n-th terms? My attempts: i think I need to ...
60
votes
11answers
2k views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
60
votes
2answers
2k views

How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$

Question: Show that $$A=\lim_{n\to \infty}\sqrt{1+\sqrt{\dfrac{1}{2}+\sqrt{\dfrac{1}{3}+\cdots+\sqrt{\dfrac{1}{n}}}}}$$ exists, and find the best estimate limit $A$. It is easy to show that ...
59
votes
7answers
6k views

What's next in this number series?

340, 680, 1428, 3141.6, _____ This is from an aptitude test. I'm not able to find any pattern in them.
58
votes
5answers
9k views

Help me solve my father's riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty. So today before leaving home he sneaked ...
53
votes
16answers
3k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, but rather grows very slowly? I'd ...
53
votes
4answers
8k views

Value of $\sum\limits_n x^n$

Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ? Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some ...
52
votes
3answers
1k views

How prove this nice limit $\lim_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$

Nice problem: Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that ...
50
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} ...
50
votes
3answers
16k views

Predicting Real Numbers

Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
48
votes
2answers
1k views

Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
48
votes
4answers
4k 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 ...
47
votes
1answer
2k views

What's the value of this Viète-style product involving the golden ratio?

One way of looking at the Viète (Viete?) product $${2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$ is as the infinite product of a series of ...
45
votes
7answers
13k views

Infinity = -1 paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
45
votes
4answers
3k views

“Closed” form for $\sum \frac{1}{n^n}$

Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...
44
votes
6answers
3k views

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
44
votes
8answers
7k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
44
votes
1answer
2k views

Does $|n^2 \cos n|$ diverge to $+\infty$?

I was recently exposed to the problem of deciding whether $$ \lim_{n \to +\infty} |n \cos n| = +\infty$$ where the limit is taken over the integers. As $|\cos n|$ oscillates throughout the interval ...
43
votes
5answers
4k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
42
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 ...
42
votes
5answers
3k views

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...
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 ...
42
votes
4answers
2k views

Double sum - Miklos Schweitzer 2010

There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is: Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
42
votes
5answers
7k views

Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly ...
42
votes
3answers
860 views

Show that $\sum{\frac{x^n}{(1+x^n)^n}}$ converges uniformly

Show that $$\sum{\frac{x^n}{(1+x^n)^n}}$$ converges uniformly on $[0,1].$ I am sorry but for this exercise I got exactly nothing. It seems to be difficult.
42
votes
0answers
897 views

Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
41
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
40
votes
7answers
895 views

Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.

Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows: $x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
39
votes
3answers
1k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\Large\frac\pi2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
39
votes
1answer
998 views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
38
votes
9answers
3k views

What is this beauty curve?

Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts. Question 1: What is the curve $f$ when $n\rightarrow \infty$? ...
38
votes
2answers
2k views

Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$

Consider the sequence defined as $x_1 = 1$ $x_{n+1} = \sin x_n$ I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
38
votes
2answers
2k views

Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?

Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge? My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here. ...
38
votes
1answer
572 views

Ways to make a series diverge “faster” to show divergence

I'm interested if there are more techniques to make a series diverge "faster" to show it diverges. Below are the specific tricks/theorems I know of to do this. I recalled reading that the sum of ...
37
votes
18answers
9k views

Getting the sequence $\{1, 0, -1, 0, 1, 0, -1, 0, \ldots\}$ without trig?

What's the simplest way to write a function that outputs the sequence: {1, 0, -1, 0, 1, 0, -1, 0, ...} ... without using any trig functions? I was able to come ...
37
votes
20answers
8k views

Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
37
votes
5answers
2k views

Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$

$$ \sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$$ Does it converge or diverge? Can we have a rigorous proof that is not probabilistic? For reference, this question is supposedly a mix of real ...
37
votes
4answers
782 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
37
votes
3answers
672 views

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...