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

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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 ...
11
votes
2answers
569 views

Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$ No clue at all. Need help! Appreciated!
15
votes
3answers
594 views

A log improper integral

Evaluate : $$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$ I found it can be simplified to $$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$ I found the exact value in the ...
14
votes
5answers
696 views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$

Evaluate $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$ I can't see anything in this limit , so help me please.
13
votes
2answers
618 views

Alternating harmonic sum $\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k$

How to analytically prove $$\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k=-\frac{11\pi^4}{360}+\frac{\ln^42-\pi^2\ln^22}{12}+2\mathrm{Li}_4\left(\frac12\right)+\frac{7\ln 2}{4}\zeta(3) $$ As O.L answer ...
2
votes
1answer
2k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
15
votes
8answers
2k views

Proving $\sum\limits_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$

Show that $$\sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$$ So for odd $n$ we have an even number of terms. So $\binom{n}{k} = \binom{n}{n-k}$ which have opposite signs. Thus the sum is $0$. For even $n$ ...
7
votes
1answer
301 views

If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$

This is a question from the book Methods of Real Analysis by R. R. Goldberg. If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: ...
5
votes
2answers
336 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
4
votes
4answers
209 views

A Convergent Series

How to prove the following equality? $$\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$
3
votes
6answers
730 views

Limit of the sequence $\lim_{n\rightarrow\infty}\sqrt[n]n$ [duplicate]

Possible Duplicate: $\lim_{n \to +\infty} n^{\frac{1}{n}} $ I know that $$\lim_{n\rightarrow\infty}\sqrt[n]n=1$$ and I can imagine that $n$ grows linearly while $n$th root compresses it ...
3
votes
1answer
540 views

Sum of tangent functions where arguments are in specific arithmetic series

By looking through an book, I found this interesting series To prove that: $$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ...
33
votes
1answer
2k views

Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$

Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$? Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question ...
21
votes
3answers
2k 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
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 ...
22
votes
3answers
939 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 ...
17
votes
5answers
948 views

Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$ Apart from this task, I asked my self: Is there a closed form for this ...
13
votes
1answer
6k views

Is the sum of sin(n)/n convergent or divergent? [duplicate]

Possible Duplicate: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ So, in my calculus class (one I'm teaching, not ...
11
votes
3answers
602 views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
5
votes
3answers
246 views

How to solve this : $\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$

How to find the sum of this : $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$$ My Working : $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} = 1-0 = 1$$ Is it ...
3
votes
3answers
303 views

Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't ...
2
votes
5answers
354 views

Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$

I am trying to find generating function for the recurrence: $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$. It looks like this: $a_0 = 1$ $a_1 = {1 \choose 2} + 3$ $a_2 = {2 ...
47
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} ...
22
votes
4answers
718 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 ...
14
votes
2answers
2k views

Limit of Nested Radical $\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots }}}$

How does one evaluate show that this limit: $$\lim_{n \to \infty}\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots \sqrt{1+(n-1) \sqrt{1+n}}}}}=3$$
17
votes
2answers
1k views

Sum of reciprocals of numbers with certain terms omitted

I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + ...
17
votes
2answers
352 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
11
votes
5answers
4k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
12
votes
4answers
430 views

Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$

I would like to show that: $$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}} $$ We have: $$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\sum_{n=0}^{\infty} ...
9
votes
3answers
718 views

Limit $\lim\limits_{n\rightarrow\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac{1}{\sqrt k}\right)$

How to find $\displaystyle\lim_{n\rightarrow\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac{1}{\sqrt k}\right)$ ? And generally does the limit of the integral of f(x) minus the sum of f(x) exist? How to ...
8
votes
3answers
11k views

Predict next number from a series

Which methods I can use to predict next number from a series of numbers ? I know the min & max possible number in advance.
7
votes
5answers
429 views

Definite integral, quotient of logarithm and polynomial

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
6
votes
2answers
883 views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : ...
6
votes
1answer
2k views

Convergence of Ratio Test implies Convergence of the Root Test

In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise: Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that ...
4
votes
3answers
3k views

Properties of $\liminf$ and $\limsup$ of sum of sequences

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n ...
7
votes
4answers
564 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
6
votes
3answers
480 views

Series as an integral (sophomore's dream)

I need help with this exercise. I need to prove $$\int_{0}^{1}x^{-x}=\sum_{n=1}^{\infty}n^{-n}$$ I think I should use some convergence theorem, but I'm stuck. Thanks a lot!
7
votes
3answers
390 views

Evaluating $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
3
votes
2answers
2k views

If a series converges, then the sequence of terms converges to $0$.

Following the guidelines suggested in this meta discussion, I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed ...
5
votes
6answers
484 views

$\sum \limits_{n=1}^{\infty}n(\frac{2}{3})^n$ Evalute Sum [duplicate]

Possible Duplicate: How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$ How can you compute the limit of $\sum \limits_{n=1}^{\infty} n(2/3)^n$ Evidently it is equal to 6 by wolfram ...
5
votes
3answers
744 views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
29
votes
5answers
757 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
19
votes
3answers
460 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
36
votes
4answers
762 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 $n = 6$; ...
20
votes
4answers
588 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: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
23
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?
14
votes
2answers
557 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
3
votes
1answer
2k views

Inequality between $\ell^p$-norms

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
12
votes
4answers
715 views

Slowing down divergence 2

Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that $ \sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)} $ diverges. (Why) must the series $$\sum_{n>1} ...
25
votes
2answers
4k 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 ...