0
votes
1answer
10 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
1
vote
1answer
13 views

Interest Accumulation - Geometric Sequence

Hello I have just worked a question in which I get an answer different to the answer in my book. The question states: If a person deposits 500 at the end of each month for 20 years at an AER of ...
1
vote
2answers
60 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
0
votes
1answer
20 views

Series and Sequences: Given $S_n = 3 n^2 - 11 n$, find $T_n$ and hence show that the series is arithmetic.

Q.Given $S_n = 3 n^2 - 11 n$, find $T_n$ and hence show that the series is arithmetic. I have attempted to solve this question. Although no luck. Can someone pleas hint me on how to find the variable ...
0
votes
3answers
24 views

Series and Sequences. In an arithmetic series T3=-2 and T9=28. How many terms of this series are required to give a sum of 1092

Q.In an arithmetic series T3=-2 and T9=28. How many terms of this series are required to give a sum of 1092. I have made an attempt at the question. By using the Sn formula, iam having trouble with ...
0
votes
1answer
35 views

Induction Proof of: Find $f(n)$ when $n=2^k$, where $f$ satisfies the recurrence relation $f(n)=f(\frac{n}2)+1$ with $f(1)=1$

How can I proof this using regular induction and strong induction? $$f(2^k)=f(\frac{2^k}2)+1=f(2^{k-1})+1$$ $$f(2^{1-1})=f(2^0)=f(1)=1$$ $$f(2^{2-1})=f(2^1)=f(2)=f(1)+1=2$$ ...
0
votes
2answers
37 views

Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence but the convergence is not uniform on $[0,1]$

Let $$ f_n(x) := \begin{cases} 1 &\text{for $x$ in } \left(0, \frac{1}{n}\right)\\ 0 &\text{$x$ elsewhere in } [0,1] \end{cases}. $$ Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence ...
0
votes
1answer
24 views

Pointwise limit,$f$, of the sequence is not bounded

Question: Let $f_n(x) := \frac{nx}{1+nx^2}$ for $x \in A := [0, \infty)$. Show that each $f_n $is bounded on $A$, but the point-wise limit of $f$ of the sequence is not bounded on $A$. Does $(f_n)$ ...
3
votes
4answers
115 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
1
vote
1answer
40 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
0
votes
2answers
21 views

A question about series ratio test

Could you please give me some hint how to deal with this question: Suppose $\left|\frac {a_{n+1}}{a_n}\right|\le c_n$ for each n and $c_n<1$. May we conclude that $\left|\frac ...
3
votes
0answers
55 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{ ...
1
vote
1answer
29 views

Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series ...
2
votes
1answer
167 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
0
votes
3answers
42 views

Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\ $$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\ $$
0
votes
1answer
28 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
0
votes
2answers
41 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
1
vote
1answer
34 views

Sequence bounds and limit

I'm doing the following exercise. Given the sequence \begin{cases} a_{n+1} = {n + 8\over4n + 1}*a_n & n=0, 1, 2 \\ a_{0} = 1 \end{cases} Find if the sequence is definitely decreasing/increasing. ...
0
votes
1answer
11 views

Multiplying non-decreasing sequences

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$). Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all ...
2
votes
1answer
31 views

Problem with Sequences..

I'm having trouble with a homework question, and cant see where im going wrong. The question is as follows: Let $x\gt1$ and let $a_n = 1+\frac 1x+\frac 1{x^2} + \cdots + \frac 1{x^n}$ for all ...
1
vote
2answers
30 views

suming an infinite series

the question involves person A eating half a loaf of bread. then person B eats half of the half left over. then person A eats half of whats left over... etc. I defined the series of person A as ...
1
vote
3answers
54 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
2
votes
2answers
55 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
0
votes
0answers
25 views

Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
0
votes
4answers
44 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
1
vote
3answers
73 views

Determine if $\sum\limits_{n=1}^{\infty}(1+\dfrac{2}{n})^n$ converges or diverges

I have an infinite series $\sum\limits_{n=1}^{\infty}(1+\frac{2}{n})^n$. I need to show if it converges or diverges using any test. I've tried applying all of the tests that I know, and hit dead ...
0
votes
2answers
40 views

A question about using Squeeze Theorem to solve theoretical convergence question

Could you give me some hint how to deal with this question: Suppose $a_n\le b_n \le c_n$ for almost all n, $b_n\to L$, $c_n-a_n\to 0$. Prove: $a_n \to L,b_n \to L$. Well, if $a_n\to a, b_n \to b$ ...
2
votes
2answers
120 views

How to calculate $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$?

Could you please help me calculate this limit: $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$. My best try is : $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac ...
1
vote
3answers
29 views

Don't know why this power series representation is wrong…

I've run into something confusing. The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$. ...
0
votes
1answer
38 views

Convergence and sum of geometric series (e^(3-2n)) as n goes from 2 to infinity

I have simplified the expression to: (e^3 / e^2n) This particular question asks to answer whether or not the series converges by virtue of |common ratio| < 1 alone, without using any other tests ...
1
vote
1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
1
vote
1answer
53 views

Is this sum converges or not?

$$\int_{n=2}^\infty \frac{\arctan\Big((-1)^nn^2\Big)}{n\ln^3n}$$ i will be glad if anyone can help me. I tried comparing it to the sum of $\Sigma_{n=2}^{\infty}\frac{1}{nlnn}$ and i said the integral ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
2
votes
2answers
86 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
2
votes
2answers
30 views

Help with Maclaurin Series

I am working on finding a Maclaurin series for this function. $$f(x) =x^6e^{x^7}$$ So I think I have to evaulate the above function based on a Maclaurin series for $e^x$ = $\sum_{n=0}^\infty ...
1
vote
2answers
77 views

Show that $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges

I'm working with some infinite series problems and I have to show that the series $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges. I don't have a lot of experience doing ...
3
votes
2answers
144 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
1
vote
1answer
35 views

Find a general formula for x_k

The sequence $x_k$... is defined by $x_0 = 0, x_1 = 2$, and $x_{k+2} = 6x_{k+1}−13x_k$ for $k≥0$. Find a general formula for $x_k$. I actually came here because I found a solution on here for a ...
0
votes
0answers
18 views

The summation of product of factorials

So the question is $\sum\limits_{x=0}^n \frac{(\beta+n-x)! (\alpha+x)!}{x!(n-x)!}$. I got the following result from mathematica yet I don't know how to prove it. Can anyone give me some help?
0
votes
0answers
30 views

Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
0
votes
0answers
32 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
0
votes
0answers
34 views

Short question on power series

After applying ratio test, the result L = x^2/4, n approaches infinity. So when you're trying to find the radius of convergence of: $$x^2/4<1$$ is it $$x<2?$$ Therefore radius is 2? I want ...
1
vote
2answers
52 views

$\sqrt[\infty]{\infty^2}$ in limit of series using root test

I'm trying to solve a problem to show if the infinite series $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{2^k}$ converges or diverges using the root test. When put in limit form, I got ...
1
vote
2answers
41 views

Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
4
votes
2answers
131 views

Infinite Series with factorial

I'm having trouble manipulating the function of this series which has factorials to show that it converges or diverges using the ratio test. The series is ...
1
vote
1answer
61 views

How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
1
vote
1answer
36 views

What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
1
vote
2answers
36 views

Sequence with an infinite amount of limit points

Find a sequence which has an infinite amount of limit points. I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle ...
0
votes
1answer
23 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
3
votes
6answers
128 views

Proving $\big(n!^{\frac1n}\big)_{n\in\mathbb N^*} \to \infty$ [duplicate]

By definition, for any $a\in\mathbb R$, there exists $k\in\mathbb N^*$ such that, if $n\in\mathbb N+k$, then $n!^{\large\frac1n}>a$. Therefore, by induction, I must: find some $k$ that satisfies ...