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

learn more… | top users | synonyms (3)

0
votes
2answers
51 views

Brian Tracy 1000% Productivity Formula?

Hi I recently watched a Brian Tracy productivity video and I am curious about the maths behind this, I understand compound interest well but I cannot seem to get the numbers working for me. How does ...
0
votes
1answer
41 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > ...
71
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: ...
1
vote
1answer
17 views

A question about Abel's test

Abel's test for $\sum a_n b_n$ to converge requires: $\sum a_n$ converges ${b_n}$ is bounded ${b_n}$ is monotone. My question is why do we need the 3d condition? The 2nd condition ...
1
vote
1answer
21 views

Find the sum of the 10th and 11th terms of the G.P.

The third term of a geometric progression of positive terms is $\frac{6}{25}$ and the seventh term is $1\frac{23}{27}$. Find the sum of the 10th and 11th terms of the G.P., giving your answers correct ...
1
vote
0answers
62 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
0
votes
0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
1
vote
0answers
47 views

Alternating Series Divergence

Test the series for convergence: $$ -2/5 + 4/6 - 6/7 + 8/8 - 10/9$$ Attempted Solution: $$a_n = (-1)^n,\; b_n = \frac{2n}{4+n}$$ $$b_n\not\stackrel{n\to\infty}{\longrightarrow}0\implies \sum_n ...
3
votes
1answer
108 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
9
votes
8answers
252 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
1
vote
1answer
36 views

Convergent strictly increasing sequence $a_n$ $\Rightarrow$ sequence $f(a_n)$ is convergent.

Strictly increasing $f(x)$ is defined on R. Then for any convergent strictly increasing sequence $a_n$ sequence $f(a_n)$ is also convergent. The answer is TRUE, but I believe I have a ...
0
votes
1answer
35 views

Is there an explicit formula for this homomorphism?

Set $A$ is a finite set of whole numbers from $1$ to $n$ (for some arbitrary $n$). Constant $c$ is an arbitrary whole number. I want to partition the set into $c$ ordered groups, so that each ordered ...
3
votes
1answer
44 views

$\sum_{n=0}^{\infty} a_n x^n$ and $\sum_{n=0}^{\infty} a_{n^2} x^n$ with different radii of convergence

Could you give an example of $$\sum_{n=0}^{\infty} a_n x^n$$ and $$\sum_{n=0}^{\infty} a_{n^2} x^n$$ that have different radii of convergence?
1
vote
0answers
28 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
14
votes
2answers
464 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
2
votes
3answers
101 views

Calc 2: convergent of divergent sequences

I would like to know if this sequence, $\sin\left(\frac{n \pi}{2}\right)$ ,is convergent or divergent? I have done this problem and I know that it is divergent through oscillation (I'm pretty sure). ...
4
votes
2answers
122 views

Limit of $a(k)$ in $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $

For n = 1, 2, 3 ... (natural number) $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $ $ a_1 = 1, \ a_2 = \frac{1}{2}, \ a_3 = \frac{7}{12} \cdots $ What is the limit of {$ a_k $} $ \lim_{k \to \infty} ...
27
votes
3answers
487 views

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following true? If $a_n$ is an integer, then $n\le 8$. I conjectured this by using ...
2
votes
2answers
97 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
0
votes
0answers
24 views

Two case about convergent series

Could you help me to prove analytically that ? I started to study Taylor Series and I'm lost. Thanks in advance.
0
votes
1answer
35 views

Solving a ODE using series

I have to prove that the series $$y(x)=\sum_{n=0}^{+\infty}\frac{x^{n}}{(n!)^{2}}$$ satisfies the ODE $$xy''+y'-y=0$$ When I derivate and substitute in the equation, I get ...
1
vote
1answer
56 views

Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
11
votes
1answer
238 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
1
vote
2answers
33 views

Test for convergence with either comparison test or limit comparison test

Tried using $b_n = \frac1{n^n + 1}$ with limit test which indicated that both either converge or diverge but getting stuck on how to show that one actually does converge.
0
votes
0answers
19 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
0
votes
3answers
41 views

Use the limit comparison test on the following series: $\sum_{n=1}^\infty\frac{5n^3+1}{2^n(n^3+n+1)}.$ [closed]

Use the limit comparison test on the following series. Can't figure out a good bn to simplify this problem. $$ \sum_{n=1}^\infty\dfrac{5n^3+1}{2^n(n^3+n+1)}.$$
2
votes
2answers
43 views

if $a>1$, Prove that $\lim a^{1\over n}=1$

if $a>1$, Prove that $\lim a^{1\over n}=1$ Is the result true if $0<a\le1 ?$ My attempt : let $a^{1/n}=1+h$, then $a=1+nh+\frac{n(n-1)h^2}{2}+\dots+h^n$ so, $a>\frac{n(n-1)h^2}{2}$ or, ...
0
votes
2answers
62 views

What number does not belong to the following series? [closed]

I was doing an IQ test with my friends, when I found a question that left me stumped: What number does not belong in this series? 2 - 3 - 6 - 7 - 8 - 14 - 15 - 30 It is only one number, and I ...
2
votes
2answers
50 views

Series convergence - Gauss test

How do I prove that $$\sum_{n=1}^\infty\left(\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot(2n)}\right)^{k}$$ converges for $k>2$ using Gauss test?
12
votes
3answers
205 views

How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $

How to show that $$ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $$ where $e = \lim \left({1 + \dfrac 1 n}\right)^n$ I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM ...
1
vote
1answer
33 views

Comparison test involiving e

Just one thing: with each sum can I compare $$a_{n}=e-\left(1+\frac{1}{n}\right)^{n}$$ to prove that the series $$\sum a_{n}$$ diverges?
1
vote
0answers
54 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
12
votes
3answers
233 views

How to prove that $ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $?

Mathematica tells me that $$ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
0
votes
2answers
36 views

Sum of this series.

I tried manipulating it to get it into a binomial expansion of two known terms, but i seemingly failed. Please help me out. $$S=\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}$$
2
votes
1answer
44 views

Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
8
votes
1answer
99 views

Interesting sum-integral equality

Is there an elementary proof of $$\lim_{n \to \infty} \int_0^\infty e^{-\alpha x^2} \frac{\sin((2n + 1)x)}{\sin x} dx = \pi\left(\frac{1}{2} + \sum_{k = 1}^\infty e^{-\alpha k^2 \pi^2}\right),$$ where ...
0
votes
1answer
14 views

Sequence of Partial Sums for repeated decimal

I have been trying to figure out an explicit formula for the sequence of partial sums of a repeating decimal. Take 0.09 repeating for example. Using the fact that it is a geometric series with r < ...
1
vote
1answer
26 views

Two cases involving Maclaurin Series

Could you help me to prove it? I'm working hard in it, but I got nothing.
2
votes
3answers
34 views

Even-Odd pair in a sequence

Suppose we have a sequence of $n$ integers not necessarily distinct. Let's define, $E$ = Number of pairs $(i, j)$ such that $i<j$ and $A_i+A_j$ is even. $O$ = Number of pairs $(i, j)$ such that ...
9
votes
2answers
136 views

How to prove that the number $1+4a_{n}a_{n+1}$ is a perfect square.

A sequence of integer $\{a_{n}\}$ is given by the conditions $a_{1}=1, a_{2}=12,a_{3}=20$,and $$a_{n+3}=2a_{n+2}+2a_{n+1}-a_{n}$$ show that for every postive integer $n$, the number ...
1
vote
3answers
31 views

Geometric Sequences

Find geometric progression if $a_1 = 3$, $S_n = 2343$, $a_n = 1875$. I'm trying to use sum formula $S_n = a_1\dfrac{1-r^n}{1-r}$, but can't do much. I'm a bit lost so if anyone could help me.
1
vote
0answers
55 views

infinity products of real numbers

are there any rules to determine if a infinity product $\prod\limits_{n\in\mathbb N}a_n$ converges? 1.it is easy to see that if $a_n>1$ for each $n$,(or $1>a_n>0$),and ...
1
vote
1answer
46 views

Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle ...
11
votes
3answers
307 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any positive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
0
votes
3answers
39 views

Arithmetic progression and right angled triangle

If the sides of a right angled triangle are in Arithmetic progression then prove that their sides are in the ratio 3:4:5 I assumed the sides as x, x+d and x+2*d and started collecting the ratios of ...
5
votes
1answer
49 views

Closed form for sequence A145271

I would like to know if there is a simple formula or method of expanding the expression given by $\left[g(x) \frac{d}{dx}\right]^n g(x)$ where $n$ is a positive integer, without having to resort to ...
3
votes
1answer
37 views

Total number of possible sub sequence with given condition

Given a sequence of two letters A and B find the total number of possible sub sequences where number of letter A is two times the number of letter B without ...
8
votes
2answers
363 views

How prove this $\sum_{n=1}^{\infty}\frac{\zeta_{2}}{n^4}=\zeta^2(3)-\frac{1}{3}\zeta(6)$

show that $$\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$$ where $$\zeta_{m}=\sum_{k=1}^{n}\dfrac{1}{k^m},\zeta(m)=\sum_{k=1}^{\infty}\dfrac{1}{k^m}$$ is true? because ...
0
votes
1answer
16 views

Multiplication of non-summable sequences convergent to $0$

Let $(\lambda_n)_n$ be a sequence of real numbers which converges to $0$ (i.e., is in $c_0$), but is not in $\ell^p$ for any $1\leq p<\infty$, e.g., $\lambda_n=\frac{1}{\log(n+2)}$ for ...
6
votes
2answers
525 views

Sum of this series

$$ \mbox{How do I find the sum of this series}\quad \sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}\ {\large ?} $$ Hints in the right direction would be appreciated.