1
vote
3answers
74 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
127 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? ...
3
votes
2answers
33 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
38 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
53 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
265 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
269 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
25 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
70 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
51 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
48 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
27 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
48 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
53 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
40 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
33 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
70 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
229 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
48 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
450 views

Find the values of p for which the series is convergent of the series …

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
85 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
74 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
55 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
29 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
51 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
70 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
73 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 ...
0
votes
2answers
40 views

What test do I use to see if the series is divergent or convergent?

$$\sum^{\infty}_{n = 1}{3n + 1 \over n^{2} - n + 2}$$ What test do I use to find out if the limit is convergent or divergent?
2
votes
3answers
71 views

Is series convergent/divergent

I need to find out is series convergent or not $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ How can I do that? Can you show step-by step solution?
1
vote
1answer
63 views

Why does this series diverge? $\sum_{n=1}^\infty \frac{n-1}{4n-1}$

So taking my original problem: $\sum_{n=1}^\infty \frac{n-1}{4n-1}$ I treated it like a limit problem as took the sum to be $\frac{1}{4}$ and since that is $<1$ for this geometric series, I ...
1
vote
1answer
31 views

Series diverges or converges

How can I show that $\sum_{n=1}^\infty (\sin n -\sin(πn/2)) / n^2$ converges or diverges? I tried using the ratio test but it's complicated I'd say it converges to $0$ since $n^2$ is growing faster ...
2
votes
2answers
98 views

Does order matter for the convergence of infinite products

Similar to infinite sums, does order matter in the convergence of infinite products? More specifically, I'm interested in the product of all rational numbers in the interval $(0,a]$. For example, ...
7
votes
2answers
106 views

Series convergence/divergence

I was trying to prove the following question. Part a is intuitive but couldn't give a clear mathematical argument. For parts b and c It seems there is something I am not seeing. Any help ? If ...
4
votes
2answers
115 views

Am I right in my conclusions about these series?

I'm trying to decide if these series converge or diverge: $$\sum_{n=1}^{\infty} (-1)^n \left(\frac{2n + 100 }{3n + 1 }\right)^n $$ Here $\lim_{n\to\infty} \left(\frac{2n + 100 }{3n + 1 }\right)^n ...
-8
votes
1answer
140 views

How to decide the convergence or divergence of these series? [closed]

What about the convergence of the following series? $$\sum_{n=1}^{\infty} (-1)^n (\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2\cdot 4 \cdot 6 \cdots (2n)})^3$$ $$ \sum_{n=1}^{\infty} (-1)^n ( 1 - n ...
6
votes
1answer
679 views

Newton-Raphson Method.

I have to find a real root of the equation $x=\ e^{-x}$ , using the Newton-Raphson Method. I solved the example writing the function in the form $f(x)=x -\ e^{-x}=0$ which is straightforward and i ...
2
votes
2answers
1k views

Fixed-Point iteration technique.

I have to find the root of $x-e^{-x}=0$ by using fixed-point iteration. when i rewrite the equation as $x=e^{-x}$ , the iterative process converges to $0.567$ after $12$ iteration. But when i ...
5
votes
4answers
153 views

Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

How can we evaluate this series $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$ Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how ...
1
vote
3answers
113 views

Does $a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$ converge?

I want to check if a sequence converges or diverges. The sequence is the following: $$a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$$ I though of maybe using sandwich theorem, but can I ...
2
votes
2answers
119 views

Is there a series $\sum (a_n) $ that converges conditionally but $\sum (a_n -1/n) $ doesn't?

I'm studying for a test in calculus and have encountered a question I can't find a proof that contradicts the existence of such series. Contradict the existence of the series such that: $\sum(a_n) $ ...
0
votes
1answer
150 views

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to (a) $0$, if $a+b<2$ (b) $2$, if $a+b>2$ ...
3
votes
1answer
102 views

hard time with series convergence or divergence [duplicate]

I'm having real hard time with this series I can't prove that the series converges and also I can't prove that the series diverges: $$\sum_{k=1}^\infty\frac{\sin^2(n)}{n}.$$ any help would be ...
2
votes
6answers
78 views

Let $p_n$ be the sequence defined by $p_n=\sum_{k=1}^n\frac{1}{k}$. Show that $p_n$ diverges even though $\lim_{n\to\infty}(p_n-p_{n-1})=0$ [duplicate]

I have tried this as : $$p_n=\sum_{k=1}^n\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}+\frac{1}{n}$$ ...
2
votes
1answer
56 views

What test should I use to classify this series?

Classify $$\sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}$$ as absolutely convergent, conditionally convergent or divergent. I am not sure whether I should use integral test or comparison test. Do you ...
1
vote
1answer
592 views

Sequence with convergent subsequences: divergent or convergent?

$(a_n)_{n \in \mathbb{N}}$ be a sequence with the convergent subsequences $(a_{2n}), (a_{2n+1})$ and $(a_{3n})$. Is $(a_n)$ then convergent? Proof or counter-example. My idea with this question was ...
2
votes
1answer
112 views

What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
3
votes
2answers
231 views

Convergence of the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$..?

Does the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$ converge..? Can anyone please give me a simple proof..