0
votes
3answers
59 views

Convergence of the series $\sum n!/(n^2+3)$

How can we test if this series diverges/converges? $$\sum_{n=1}^\infty\frac{n!}{n^2+3}$$ I tried D'Alembert's principle and tried to do $\frac{a_{n+1}}{a_n}$ but I'm stuck. Any help?
0
votes
1answer
74 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ [on hold]

I need to find out whether this series converges or diverges. $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ Can someone help how to solve it?
1
vote
2answers
31 views

Convergence test of certain series

I need to find out whether this sequence converges or diverges using limit comparison test. $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$ I've tried it with the use of sequence ...
3
votes
1answer
67 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} = ...
1
vote
3answers
89 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
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
106 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
votes
2answers
70 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [closed]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
2
votes
2answers
84 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
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 ...
1
vote
1answer
27 views

Two cases involving Maclaurin Series

Could you help me to prove it? I'm working hard in it, but I got nothing.
1
vote
0answers
49 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
0
votes
1answer
60 views

Concerning the sum $\sum_{n = 1}^\infty \sin nx$

I recently came across this question and I posted an answer. It has been pointed out that my answer is incorrect. I cannot work out what is wrong with my reasoning. The answer I gave corresponds with ...
1
vote
2answers
52 views

Divergence of modified harmonic series

I am reading a paper which claims that the following series diverges: $\sum\limits_{n=2}^{\infty}\frac{1}{nH_{n-1}}$ where $H_{n}$ is the $n$'th harmonic number $\sum\limits_{m=1}^{n}\frac{1}{m}$. I ...
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 ...
1
vote
2answers
32 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
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 $0<a_n<1$, $n=1, 2, 3\dots$ ?
38
votes
1answer
981 views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
-4
votes
1answer
33 views

Question on convergent and divergent sequences [closed]

Is every divergent sequence constant? Please provide an example. Thanks!
0
votes
1answer
22 views

Evaluate the following infinite series or state that the series diverges.

From my textbook. $$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$ My work: So a constant greater than or equal to $1$ raised to ∞ is ∞. A number $n$ for $0<n<1$ is $0$. So when taking the ...
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
2answers
71 views

Prove this series is divergent: $\sum_{k=1}^{\infty}\sin kx$

I need to prove that $\sum_{k=1}^{\infty}\sin kx$ is divergent when $x \notin \pi \Bbb Z$. I tried to solve this equation with it's sums' seria but I didn't succeed. I'll be glad for some help.
0
votes
4answers
51 views

how to find out series is divergent or convergent for $\sum_{n=1}^\infty \frac{2^n}{n^2}$

\begin{equation}\sum_{n=1}^\infty \frac{2^n}{n^2}\end{equation} The text book says the above series diverges by the n-th term test, but given no procedures how it was done so, could you some ...
1
vote
3answers
46 views

$\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent..

I am stuck on the following problem that says: Let $\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent. Then it follows that a. $a_n \to 0$ but ...
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|}) ...
1
vote
1answer
58 views

Value of divergent series?

Let $\{a_n\}$ be a positive, convergent sequence. We consider the sequence of partial sum $\{s_n\}: s_n = \sum_{k=1}^n a_n$. Clearly $\{s_n\}$ is strictly increasing and therefore $\sum_{n=1}^\infty ...
1
vote
1answer
20 views

Limit Comparison Question

I have a interesting problem in my book. It states: Show that if $a_n > 0$ and $\lim\limits_{n\to\infty} (n \cdot a_n) \neq 0$, then $\sum a_n$ is divergent. It hints at using limit comparison ...
0
votes
2answers
33 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
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
40 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 ...
0
votes
1answer
60 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
0
votes
3answers
63 views

How does this series diverge?

The series: $$\sum_{n=0}^{\infty} \sqrt{n^2 +1} -n$$ diverges. Can someone please tell me how this is proven and done.
0
votes
2answers
33 views

Determining Divergence

How can I prove that this series diverges? I don't think you can use a comparison test, but maybe I'm mistaken. $$\sum \dfrac 1{n^{4/5}+10^{10}}$$
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
199 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 ...
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 ...
7
votes
0answers
132 views

How find this series $\sum_{n=1}^{\infty}\frac{1}{n^2H_{n}}$?

Question: This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$ I suddenly thought of this ...
1
vote
3answers
101 views

Divergent Alternating Series

Need help in finding an alternating series: S = $\sum_{n=1}^{\infty}(-1)^{n+1}b_n$ where $\lim_{n\to \infty}b_n = 0$ $b_n > 0$ but only $\forall n \ge 1$ such that S diverges
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 ...
2
votes
1answer
49 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
-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 $ ...
3
votes
0answers
44 views

Is there an infinite hierarchy of sequences whose reciprocals diverge, starting with the natural numbers?

It is well known that the sum of the reciprocals of the function $f_0(n)=n$ (the harmonic series) diverges: $$\sum_{n=1}^\infty\,\frac{1}{n}=\infty$$ Similarly, the sum of the reciprocals of the ...
1
vote
0answers
32 views

Naive calculations with infinite series [duplicate]

In the realm, where the sum of natural numbers is $-1/12$ : $1+2+3+4+...=-1/12$ Is this true?: $2+4+6+8+...=2*(1+2+3+4+...)=-2/12$ Can this kind of naive calculations always be done? -or are there ...
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. ...