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

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12
votes
2answers
595 views

Double sum trouble

Evaluate: $$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}$$ Honestly, I don't see where to start with this. I am sure that this is a trick question and I am missing something ...
8
votes
4answers
634 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 ^^
10
votes
2answers
2k views

Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?

Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges. I calculated $$ ...
10
votes
1answer
528 views

Proving that $ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+…+\frac{1}{\sin(133°)\sin(134°)}=\frac{1}{\sin(1°)}$

I would like to show that the following trigonometric sum $$ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+\cdots+\frac{1}{\sin(133°)\sin(134°)}$$ ...
8
votes
4answers
425 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 ...
7
votes
5answers
922 views

Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
6
votes
2answers
610 views

Convergent series exercise from Little Rudin

Problem: Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that $$\sum_0^\infty a_n$$ converges, show that $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$ ...
5
votes
1answer
1k 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 ...
15
votes
4answers
568 views

Generalisation of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$

Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ? For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than ...
10
votes
4answers
193 views

Convergence of $\frac{\sqrt{a_{n}}}{n}$

Can anyone help me with the following question. If $a_{n} \geq 0$ and $\sum a_{n}$ converges then how to prove $\sum \frac{\sqrt{a_{n}}}{n}$ converges. Any idea where to start. My idea was to try ...
9
votes
2answers
2k views

Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $[0,2\pi]$

This question arises because of a problem I was doing (Bartle 3rd edition, section 9.4 problem 3). It was like this. Given $a_n$ a decreasing sequence of positive numbers and suppose that ...
8
votes
3answers
321 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 ...
7
votes
1answer
317 views

Convergence of series involving in iterated logarithms $\sum \frac{1}{n(\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k} }$

What is the quickest way to show when $$ S(\alpha_1,\alpha_2,\cdots,\alpha_k) = \sum\limits_{n=3}^\infty \frac{1}{n (\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k}} $$ converges, where $\log^k(n)$ ...
7
votes
2answers
373 views

How can I prove Infinitesimal Limit

how can I prove this trouble: Prove that if $$\lim_{x\to 0} f(x) = 0,$$ and $$\lim_{x\to 0} \frac{f(2x)-f(x)}{x}= 0,$$ then, $$\lim_{x\to 0} \frac{f(x)}{x} = 0.$$ i try to solve it in this way: ...
5
votes
4answers
160 views

Proof of $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$

Why would $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$ be true where $(a_n)$ is a sequence in $\mathbb{R}$? Edit: Let all $a_n$ be positive.
5
votes
2answers
930 views

Is there a common symbol for concatenating two (finite) sequences?

Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = ...
4
votes
4answers
612 views

Show that this limit is equal to $\liminf a_{n}^{1/n}$ for positive terms.

Show that if $a_{n}$ is a sequence of positive terms such that $\lim\limits_{n\to\infty} (a_{n+1}/a_n) $ exists, then this limit is equal to $\liminf\limits_{n\to\infty} a_n^{1/n}$. I am not ...
3
votes
1answer
544 views

Can you please check my Cesaro means proof

I wanted to prove the following: if $x_n \to x$ then $y_n \to x$ where $$ y_n = {x_1 + \dots + x_n \over n}$$ Please can you tell me if my proof is correct? My proof is this: Let $\varepsilon > ...
12
votes
2answers
2k views

Do these series converge to the Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
7
votes
5answers
867 views

Calculate the sum of the infinite series $\sum_{n=0}^{\infty} \frac{n}{4^n}$

A previous problem had us solving $\sum_{n=0}^{\infty} \frac{1}{4^n}$ which I calculated to be $\frac{4}{3}$ using a bit of mathematical manipulation. Wonderful. Thank you for all the prompt ...
6
votes
2answers
183 views

Checking my understanding: $1 - 1 + 1 - 1 + 1 - … = \frac{1}{2}$

I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows: Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...
6
votes
3answers
252 views

The series expansion of $\frac{1}{\sqrt{e^{x}-1}}$ at $x=0$

The function $ \displaystyle\frac{1}{\sqrt{e^{x}-1}}$ doesn't have a Laurent expansion at $x=0$. But according to Wolfram Alpha, it does have a series expansion that includes terms raised to ...
6
votes
4answers
883 views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
4
votes
1answer
807 views

Sum of product of Fibonacci numbers

I want to calculate the sum of product of Fibonacci number for a given $n$. That is, for given $n$, say $$F_1 F_n + F_2 F_{n-1} + F_3 F_{n-2} + F_4 F_{n-3} + F_5 F_{n-4} + \cdots.$$ what should be ...
4
votes
2answers
48k views

Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
3
votes
1answer
336 views

Evaluating the time average over energy

For more info see the article equations 37 Edit: The $\varepsilon ^3 $ has vanished due to time average. But how to get the 4th order? Let us define some function for scalar field $$\phi= ...
2
votes
4answers
340 views

can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer

derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$ observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer $t_{0}=0$
2
votes
5answers
221 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 ...
2
votes
2answers
414 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
2
votes
6answers
378 views

Sequence sum question

I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$ Can anybody help me?
6
votes
2answers
310 views

Checking of a solution to How to show that $\lim \sup a_nb_n=ab$

In course of solving the problem How to show that $\lim \sup a_nb_n=ab$ I feel that I've probably made some mistake in my solution for I didn't use the fact that $a_n>0$ $\forall$ $n\geq1.$ ...
5
votes
3answers
664 views

Limit of a particular variety of infinite product/series

I was musing about a particular limit, $L = \prod\limits_{n > 0} \bigl(1 - 2^{-n}\bigr)$: we may bound 0.288 < L < 0.308, which we may show by taking the logarithm: ...
2
votes
1answer
135 views

Elementary proof, convergence of a linear combination of convergent series

Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also ...
-8
votes
5answers
942 views

How does the number $ e $ relate to the limit of $(1+1/n)^n$ as $n \to \infty$? [closed]

The number $e$ is sometimes defined as $\lim_{n\to\infty}\left(\frac{n+1}{n}\right)^n$. Does this mean we can say that $e = \left(\frac{\infty + 1}{\infty}\right)^{\infty}$? Why or why not? And what ...
44
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 ...
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 ...
19
votes
4answers
505 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
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 ...
26
votes
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 ...
13
votes
2answers
685 views

Sum of the squares of the reciprocals of the fixed points of the tangent function

The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...
27
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
13
votes
9answers
2k 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, ...
11
votes
1answer
298 views

Forcing series convergence

I am trying to figure this out: $\mathscr{S}=\big\{(a_n),(b_n),\dots \big\}$ is a finite set of real, null sequences. Does there exist a sequence $(\epsilon_n)$, where $\epsilon_k=\pm 1$ for each ...
13
votes
3answers
323 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
10
votes
2answers
465 views

How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$

How can I evaluate the following series: $$1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}+\frac{3\cdot 5\cdot 7\cdot 9}{4\cdot 8\cdot 12\cdot 16}+\cdots$$ In one ...
8
votes
2answers
469 views

Proving a function is continuous on all irrational numbers

Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb ...
4
votes
4answers
9k views

relation between integral and summation

What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here where I got two answers (one is based on summation notation) and the ...
11
votes
2answers
351 views

How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?

It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
8
votes
1answer
389 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 ...
7
votes
2answers
714 views

Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$

We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$