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

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23
votes
4answers
732 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: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
19
votes
4answers
2k 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. ...
7
votes
7answers
4k views

Series that converge to $\pi$ quickly

I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
15
votes
4answers
881 views

Where do summation formulas come from?

It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is ...
10
votes
1answer
518 views

The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent

If a series $\sum\limits_{n=1}^\infty a_n$ is convergent, and $a_n\gt0$... Do not refer to Carleman's inequality or Hardy's inequality, show that the series $$\sum_{n=1}^\infty \frac ...
14
votes
4answers
799 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} ...
12
votes
5answers
663 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 ...
6
votes
5answers
643 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$ [duplicate]

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
16
votes
3answers
904 views

How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
15
votes
2answers
2k 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
7answers
12k views

Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?

Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$? Can someone point me to a proof, or explain if it's a simple answer? What I'm looking for is the point where it becomes understood that trigonometric ...
27
votes
2answers
8k 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 ...
11
votes
3answers
474 views

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...
9
votes
2answers
1k views

Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?

I've had no luck with this one. None of the convergence tests pop into mind. I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I ...
8
votes
4answers
710 views

Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$

For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$ how can I find the limit of sequence above? Provide me a hint or full solution. thanks ^^
6
votes
3answers
580 views

Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$

How does one sum the series $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$ This was asked to me by a high school ...
19
votes
3answers
1k 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 ...
10
votes
2answers
567 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
5
votes
3answers
418 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 ...
5
votes
1answer
2k views

Fly and Two Trains Riddle

Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the ...
3
votes
6answers
1k views

Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $ [duplicate]

Possible Duplicates: Finding the limit of $n/\sqrt[n]{n!}$ How come such different methods result in the same number, $e$? I've seen this formula several thousand times: $$e=\lim_{x\to ...
8
votes
3answers
351 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 ...
8
votes
1answer
1k views

Closed form of a partial sum of the power series of exp(x)

I am looking for a closed form (ideally expressed as elementary functions) of the function $\exp_n(x) = \sum_{k=0}^n x^k / k!$. I am already aware of expressing it in terms of the gamma function. ...
7
votes
3answers
7k views

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

$$1 + \sum_{j=1}^{n} j!j$$ 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 ...
8
votes
5answers
627 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but ...
3
votes
4answers
9k views

Is the sum of all natural numbers $-\frac{1}{12}$? [duplicate]

My friend showed me this youtube video in which the speakers present a line of reasoning as to why $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ My reasoning, however, tells me that the previous ...
3
votes
1answer
800 views

absolutely convergent series and conditionally convergent series rearrangement

I don't understand that why the terms of an absolutely convergent series can be rearranged in any order and all such rearranged series converge to the same sum. my textbook gives me an example that ...
3
votes
3answers
9k views

Proving formula for product of first n odd numbers

I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$): $$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$ How ...
2
votes
5answers
231 views

Summation equation for $2^{x-1}$

Since everyone freaked out, I made the variables are the same. $$ \sum_{x=1}^{n} 2^{x-1} $$ I've been trying to find this for a while. I tried the usually geometric equation (Here) but I couldn't get ...
1
vote
3answers
2k views

limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
76
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.
36
votes
5answers
1k views

A question on Taylor Series and polynomial

Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$. For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial. Is true that $f(x)$ is a ...
29
votes
6answers
2k 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 ...
25
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? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
34
votes
3answers
1k views

Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$

Can be easily proved that the following series onverges/diverges? $$\sum_{k=1}^{\infty} \frac{\tan(k)}{k}$$ I'd really appreciate your support on this problem. I'm looking for some easy proof here. ...
46
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 ...
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
28
votes
2answers
1k 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 & ...
17
votes
9answers
3k 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, ...
17
votes
4answers
1k views

The sum of $(-1)^n \frac{\ln n}{n}$

I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$ This is a problem in Calculus by Simmons. It's in the end of chapter review and it's ...
17
votes
2answers
1k views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
13
votes
1answer
539 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
13
votes
4answers
944 views

Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$

We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges. One can then ask the question of ...
28
votes
1answer
775 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: ...
12
votes
3answers
360 views

Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$

Let $\sum a_{n}$ be an absolutely convergent series such that $$\sum a_{kn}=0$$ for all $k\geq 1$. Help me prove that $a_{n}=0$ for all $n$. Thank you!
12
votes
4answers
746 views

How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?

I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me?
8
votes
1answer
115 views

Sequence Limit Problem: If $0 \leq x_{m+n} \leq x_n + x_m$ then limit of $x_n/n$ exists

If the sequence $\{x_n\}$ satisfies the property that $0 \leq x_{m+n} \leq x_n + x_m$ for all $n$, $m \in \mathbb{N}$ , show that the limit of the sequence $\left\{\frac{x_n}{n}\right\}_n$ exists. ...
8
votes
5answers
406 views

Finding sum of a series: difference of cubes

I am trying to find sum of the infinite series: $$1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots$$ I tried to solve it by subtracting sum of even cubes from odd, but that solves only half of the ...
8
votes
2answers
2k views

Sufficient condition for convergence of a real sequence

Let $(x_n)$ be a sequence of real numbers. Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the ...
6
votes
3answers
417 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. ...