3
votes
1answer
65 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
3
votes
5answers
133 views

Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
3
votes
3answers
105 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
1
vote
2answers
87 views

How to tell if a log series converges?

I have the following series. $$(-1)^n \times \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)$$ I tried the root, ratio and integral tests, but am doing something wrong because I am unable to tell if this series ...
2
votes
2answers
41 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
5
votes
4answers
113 views

Is $\sum\limits_{n=1}^\infty \sin{\frac{(-1)^{n+1}}{n}}$ convergent?

$$ \mbox{Is}\quad \sum_{n=1}^\infty \sin\left(\left[-1\right]^{n + 1} \over n\right) \quad\mbox{ convergent ?.} $$ $$ \mbox{I know that }\quad\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\quad \mbox{is ...
0
votes
1answer
27 views

$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
0
votes
1answer
43 views

Prove or disprove the convergence of…

I need help with the following problem, please help. For positive real x. Let $${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$ Prove or disprove the convergence ...
0
votes
1answer
78 views

Convergence of $\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$

$\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$ A) For which $p\in \mathbb{R}$ is the series convergent? B) For which $p\in \mathbb{R}$ is the series divergent, and what is ...
19
votes
1answer
257 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
0
votes
1answer
31 views

Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
-4
votes
1answer
33 views

Question on convergent and divergent sequences [closed]

Is every divergent sequence constant? Please provide an example. Thanks!
1
vote
1answer
62 views

Convergence or Divergence of a Series Using Case Analysis

In the problems below it's asked for which $r \in \Bbb R$ the series converges. $$ a)\quad\sum_{k=0}^\infty \left( \left(\sum_{l=1}^k \frac1l\right) r^k\right) $$ $$ b)\quad\sum_{k=0}^\infty ...
0
votes
1answer
15 views

What convergence test can I use on this series?

I am doing practice problems for an exam, and I am not sure how to test this series: Limit from n=1 to infinity of cos(n) * sin^2(1/n) I am supposed to use lim x -> 0 sin(x)/x = 1 to find the ...
0
votes
2answers
31 views

Find any sequence that meets these criteria.

I'm struggling with this problem and don't know where to start looking: Is there any sequence $a_n$ such that $\lim\limits_{n \to \infty}a_n \neq 0$ and $\lim\limits_{n \to \infty}(n \sqrt[n]{|a_n|}) ...
0
votes
2answers
24 views

Power Series — Convergence, Divergence, and Absolute Convergence

Suppose that the power series $$\sum a_nx^n$$ is convergent at $x=-3$ and divergent at $x=5$. What can be said about the following: convergence at $x=-2$ ? absolute convergence at $x=2$ ? ...
1
vote
1answer
44 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
1
vote
4answers
101 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
3
votes
2answers
28 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
0
votes
1answer
38 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
1
vote
3answers
90 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
0
votes
4answers
198 views

Why do some series converge and others diverge? [closed]

Why do some series converge and others diverge? What causes the divergence or convergence of a series and why does that cause such a behavior? For example, why does the harmonic series diverge, but ...
4
votes
1answer
37 views

Divergence of $\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}…$

Assume that we know this converges. $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+....$$ Is it possible to detect for which largest $0<s<1$ the sum below diverges? ...
5
votes
2answers
42 views

Help with Convergence/Divergence

So I am trying to prove whether the following problem converges or diverges? $$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$ So I decided to use the Root test. $$ L = \lim_{n\to ...
1
vote
2answers
43 views

Does it converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get 1. I suppose I should compare it with some other series, but I can't figure out with ...
-1
votes
2answers
55 views

Determine whether the following series convergent? [closed]

Is the following series convergent? $$\sum_{k = 1}^{\infty}{1 \over k\left[1 + \ln^{2}\left(k\right)\right]}$$
5
votes
4answers
288 views

Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$
2
votes
1answer
342 views

The series $a_n$ is conditionally convergent. Prove that the series $n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0 $ ...
1
vote
1answer
30 views

Determining convergence/divergence

I am studying for my exam tomorrow and have come across some problems I cannot get. I have put them below with what I have tried/thinking process behind. Thank you. ...
0
votes
4answers
81 views

Does the following series converge or diverge?

(2n-1)/(n!) I used the ratio test here and got the lim as n -> infinity to be 0. Therefore, I assumed that the series converges. However, my textbook says that it ...
2
votes
0answers
53 views

Showing how this infinite sum diverges [duplicate]

$\displaystyle{\sum_{n=1}^{\infty} \left[\left( 1 + {1 \over n}\right)^{n} - {\rm e}\right]}$ I tried both root and ratio tests (for the root test, the expression became way too complex to handle) ...
0
votes
2answers
54 views

Does the Leibniz Test say that if the sequence doesn't go to 0 that the series diverges?

My book explains the Leibniz test by saying: "Assume a sub n is a positive sequence that converges to 0..." And goes on to say that that means the alternating series converges. What if the sequence ...
0
votes
0answers
34 views

Prove convergence of a serie little with Direct comparison test

I have the following Serie $$\sum_{k=1}^{\infty} \log(1+\frac{1}{k^2})$$ this serie should converge but when i apply the Direct comparison test it diverges $$|\sum_{k=1}^{\infty} ...
-1
votes
1answer
65 views

Convergence, Divergence and Summability of this series

If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? $\sum_{n=0}^{\infty} ...
2
votes
3answers
54 views

Series convergence problem

I´m having trouble analyzing the convergence of another series: $$\sum_{n=1}^\infty (-1)^n \frac 1 {n^{\frac 1 n}}$$ If you could just give me a hint about which test for convergence should I use, ...
0
votes
3answers
43 views

Convergence of an infinite series problem

I am having trouble with the series $$\sum_{n=1}^\infty (-1)^n\frac n {n+1}$$ I want to know if it converges or not, and I´ve tried with the comparision test, the ratio test, the Leibniz test... ...
0
votes
2answers
38 views

Does not converge nor does it diverge to infinity or negative infinity

I am stuck on part D of my problem. Suppose that $a_n$ converges to $0$ and $b_n$ converges to infinity. $c_n = (a_n) \times (b_n)$, Give an example where $(c_n)$ does not converge, nor does it ...
1
vote
4answers
75 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
0
votes
1answer
56 views

Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \ $, $ z \in \mathbb{C}$

I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$ If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \ $?
3
votes
4answers
237 views

$2 - 4 + 6 - 8 + \cdots = 1 - 1 + 1 -1 + \cdots$?

I want to talk about the weirdness of $2\sum\limits_{n=1}^\infty n(-1)^{n-1}$ , $$\sum\limits_{n=1}^\infty n(-1)^{n-1} = 1 - 2 + 3 - 4 + 5 - 6 + \dots$$ $$\times 2 \implies 2\sum\limits_{n=1}^\infty ...
0
votes
2answers
57 views

Find values of $p$ for which the series is convergent.

I am still having a hard time understanding how to find what values of $p$ allow for the series to converge. Here is the series I am currently trying to find $p$ for: $$\sum_{n=1}^\infty n(1+n^2)^p$$ ...
1
vote
1answer
1k views

Find the values of p for which $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent

Find the values of p for which the series $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent. I know that the p-series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent if $p>1$ and divergent if ...
1
vote
2answers
91 views

Is the series $\sum_{n=1}^\infty \frac{1}{n^3+n}$ convergent or divergent? [closed]

Is the series $\sum_{n=1}^\infty \frac{1}{n^3+n}$ convergent or divergent? And how? What method is required? Thanks.
0
votes
2answers
83 views

Does the series from $1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge?

Does the series from $n = 1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge? Steps/tips would be greatly appreciated. Thanks!
1
vote
0answers
56 views

Behavior of $\sum_{k=a}^{b} {\frac{1}{k(\log k)(\log \log k)\cdots(\log^n k)}}$ under limits.

Define $$S_n (a, b)=\sum_{k=a}^{b} {\frac{1}{k(\log k)(\log \log k)\cdots(\log^n k)}}$$ where $\log^n$ denotes $n$ compositions of the natural logarithm. And $$P(n>2)=\lfloor ...
1
vote
0answers
34 views

Is this series of products convergent/divergent

Please Help I've this series: $ \sum_{p=1}^q \prod_{k=p}^q \dfrac{n-k}{k}.\dfrac{1}{p(n-p)^2} $ with $1 \leq q \leq n/2$. Is this convergent or divergent? Many thanks!
2
votes
3answers
61 views

Check the convergence of series

$$\sum _{n=1}^{\infty } \frac{\left(2 n^2-n+1\right)!}{3^{n^2+1}}$$ Trying to solve with sign d'Alembert, nothing comes out, and prevents the transformation of quadratic factorial reduction. Wolfram ...
2
votes
1answer
88 views

Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

I have to check for which $x$ the series converges/diverges. $\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$ I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > ...
1
vote
3answers
79 views

Do I use partial fractions? How to set it up?

$$\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+2}\right)$$
0
votes
2answers
38 views

What test is this am I missing something?

$$\sum^{\infty}_{n = 0}{5n^{3} \over 2n^{3} +1}$$ $$ {5n^{3} \over 2n^{3} +1}<$$$${5n^{3} \over 2n^{3} } <$$ $$ {5 \over {2} }\ne 0$$ so the series is divergent but what test is this? am I ...