1
vote
1answer
59 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
1answer
26 views

Convergence parameter: Find the value of $p>0$ for which the series converge

For the sum for $k=2$ to infinity: $$\frac{\ln k}{k^p}\ $$ The textbook says the answer is $p>1$.
2
votes
1answer
37 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
0
votes
1answer
28 views

Use comparison or limit comparison test to determine whether the series converge [on hold]

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
2
votes
1answer
48 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
3
votes
3answers
117 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
2
votes
2answers
77 views

Convergence of $ \sum_{n=1}^{\infty} (\frac{n^2+1}{n^2+n+1})^{n^2}$

Find if the following series converge: $$\displaystyle \sum_{n=1}^{\infty} \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ What I did: $$a_n=\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ $$b_n=\frac ...
1
vote
1answer
28 views

Let $ \sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ \sum_{n=1}^{\infty}2(a_n)^3$ converges as well

Let $ S_1=\displaystyle\sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ S_2=\displaystyle\sum_{n=1}^{\infty}2(a_n)^3$ converges as well. We have $\exists l :\forall ...
1
vote
2answers
64 views

The limit of $\ln(n) - \ln(n^2 + 1)$ as $n\to\infty$

As $n\to\infty$, what is the limit of $\ln(n) - \ln(n^2 + 1)$ Using properties of logs and limits, I ended up with: $$ \ln \left(\lim \left(\frac{n}{n^2 + 1}\right)\right) $$ where lim is the ...
1
vote
1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
0
votes
0answers
28 views

Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
1
vote
1answer
43 views

Does the sequence converge? To what?

Let α and β be positive real numbers and define a sequence by setting $s_1 = \alpha, s_2 = \beta$ and $s_{n+2} = \frac12(s_n+s_{n+1})\forall n\in \Bbb \ge1$ Show that the subsequences $\{s_{2n}\}$ ...
4
votes
2answers
120 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
1
vote
1answer
34 views

Understanding pointwise convergence vs. uniform convergence example

I'm trying to understand the difference between pointwise convergence and uniform convergence. I read this post and the last answer on it is the following: $f_n\to f$ pointwise on $(a,b)$ if for ...
3
votes
1answer
43 views

Alternating series limit question [closed]

Suppose $b_n > 0$ for all $n\geq1$ and $$\lim_{n\to\infty}n\left(\frac{b_n}{b_{n+1}}-1\right)>0,$$ show that the alternating series $\sum_{n=1}^\infty(-1)^n b_n$ converges.
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. ...
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 ...
3
votes
5answers
387 views

What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
0
votes
2answers
61 views

Show that $\lim_{n \to \infty} a_n b_n = ab$?

alright so I have to assume that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$ And then I have to show that $\lim_{n\to\infty} a_n \ b_n=ab$ by using the definition of convergence of ...
1
vote
1answer
34 views

Alternating series, does the series converge or diverge?

Does the series $\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{\sqrt [3]{n+1}}}$ converge or diverge? The series can be written as $\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ...
0
votes
3answers
60 views

How to prove the convergence or non-convergence of $\displaystyle \sum_{-\infty}^{\infty}\frac{n^{13}}{n^{13.2}+2n^6-1}$

My problem is to prove that the following series is convergent or non-convergent: $$\displaystyle \sum_{-\infty}^{\infty}\frac{n^{13}}{n^{13.2}+2n^6-1},\;\;\;\;\;n=\pm 1, 2, 3, ...$$ Any hints how ...
1
vote
0answers
30 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
0
votes
3answers
23 views

Convergence / absolute convergence of alternating infinite series?

How can it be shown that $\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{\sqrt [3]{n+1}}}$ is convergent and /or absolute convergent?
0
votes
0answers
46 views

When the series $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$ converge and diverge

How to, depending on the real parameters a and b, determine when will the series converge and when will it diverge: $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$. I got this for homework ...
5
votes
4answers
267 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: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$
0
votes
3answers
90 views

To be proved that $\lim \limits_{n \to \infty} n^{\frac{1}{n}} = 1$ [closed]

I need to show / prove that the $\lim \limits_{n \to \infty} n^{1/n}= 1$.
2
votes
2answers
101 views

Does the series $\sum\limits_{n=1}^{\infty}n\tan\left(\frac {\pi}{2^{n+1}}\right)$ converge or diverge?

Does the series $\sum\limits_{n=1}^{ \infty}n\tan\left( \dfrac { \pi}{2^{n+1}}\right)$ converge or diverge? My idea was to use the limit comparison test and $\sum\limits_{n=1}^{\infty} \dfrac ...
1
vote
2answers
37 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
2
votes
1answer
271 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
votes
1answer
99 views

Does the series converge or diverge as n-> inf: cos(n)/n^3

It's pretty obvious that it converges, seeing as n^3 continues getting larger, and cos(n) is bounded by 1 and -1. The ratio and root tests are useless. I was just wondering if I could use the ...
-1
votes
0answers
78 views

How to get the sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\left(1+\frac1n\right)^n-\frac{e}{2n}\right\}$??

How can we find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\left(1+\frac1n\right)^n-\frac{e}{2n}\right\}$? I accidentally saw a question to get the value of ...
1
vote
2answers
101 views

For which $\alpha$ does the series $\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}}\!\! - \!1\big)$ converge?

For what $\alpha$, does $\displaystyle\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}} - 1\big)$ converge? Divergence of $\sum_{n = 1}^{\infty}(2^\frac{1}{n} - 1)$ prompted this question.
1
vote
4answers
175 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
0
votes
1answer
72 views

Discuss the convergence of the series $\sum_{n=2}^\infty(\ln{n})^{-\ln(\ln{n})}$

Discuss the convergence or divergence of the series where the general term $x_n$ is given by $$x_n=(\ln{n})^{-\ln(\ln{n})}.$$
0
votes
1answer
61 views

Improper integral convergence from minus to positive infinity

Quote from Essential Calculus: Early Transcendentals, by James Stewart: If $f$ is continuous, then $$\int_{-\infty}^\infty f(x)dx=\lim_{t \to \infty}\int_{-t}^tf(x)dx$$ I thought this would be ...
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 ...
1
vote
1answer
47 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
1
vote
2answers
35 views

Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
1
vote
2answers
40 views

Show that If $\sum_{n=1}^\infty b_{n}$ is abosolutley convergent, then $|\sum_{n=1}^\infty b_{n}| \leq | \sum_{n=1}^\infty |b_{n}|$

this problem was given as a practice problem for first year calculus class. Here it is: show that if the series $\sum_{n=1}^\infty b_{n}$ is abosolutley convergent, then $|\sum_{n=1}^\infty b_{n}| ...
1
vote
2answers
41 views

Prove that if $\int_a^\infty g(x) dx$ is convergent then $\int_a^\infty f(x) dx$ is convergent.

where $f$ and $g$ are positive continuous functions on $[a, \infty)$, and $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = 0$$ I tried to prove this as follows: But something tells me I can't do the ...
4
votes
2answers
274 views

If $x>1$, then $\lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

How can I prove that $$ \lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x, $$ whenever $x>1$. Here $\left\lfloor y\right\rfloor$ denotes the floor ...
1
vote
1answer
27 views

Determining a series convergence using root test

I just have a few quick questions on using the root test to determine the convergence a series. If I get a series $\sum_{n=5}^\infty A_n$. What would the $n=5$ do? From my knowledge when I apply root ...
2
votes
2answers
76 views

Prove that if $\sum_{n=1}^{\infty} |a_n|$ converges and $(b_n)^{\infty}_{n=1}$ is a bounded sequence, then $\sum_{n=1}^{\infty} |a_nb_n|$ converges

this was given as an exercise: Prove that if $\sum_{n=1}^{\infty} |a_n|$ converges and $(b_n)^{\infty}_{n=1}$ is a bounded sequence, then $\sum_{n=1}^{\infty} |a_nb_n|$ converges This is what i was ...
0
votes
1answer
41 views

Prove that if $\sum_{n=1}^\infty a_{n}$ is absolutely convergent, then $|\sum_{n=1}^\infty a_{n}| \leq | \sum_{n=1}^\infty |a_{n}|$

Hey everyone this was given as a practice problem for my first year calculus class and it really giving me a headache, any help is appreciated! Prove that if $\sum_{n=1}^\infty a_{n}$ is absoultley ...
21
votes
2answers
737 views

Find the value of the sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

How can we find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\big(1+\frac1n\big)^n\right\}$? This problem appears in: T. Andreescu, T. Radulescu & V. ...
2
votes
1answer
33 views

Behaviour of generating function for distinct partitions

We all know that the generating function for distinct partitions is $$Q(x)=\prod\limits_{k=1}^{\infty}(1+x^k)$$ Computation on Maple suggests that $\lim\limits_{x\to1^-}Q(-x)=0$, $\lim\limits_{x\to ...
6
votes
0answers
83 views

Convergence of $\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$

Does $\displaystyle\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$ converges ? Let's call the inner sum $a_k$ such that $\displaystyle\sum_{k=1}^{\infty} (a_k)^{-k}$, applying ...
0
votes
2answers
75 views

Question about convergence of $\sum a_n \cdot b_n$ when $\sum a_n$ converges.

So i have some series, it is $\sum a_n$ and i know it converges absolutely. Is it true that for given: $$b_n = \frac{n^2+1}{n^2}$$ $\sum a_n \cdot b_n$ converges absolutely too? What about $\sum ...
1
vote
0answers
38 views

Convergence set of power series

I am trying to find the convergence set of the power series: $\sum_{n=1}^\infty ln\big[1+\big(\dfrac{1}{n}\big)\big](x+2)^n$. So using the ratio test: $\lim_{n\to\infty} \dfrac{|a_{n+1}|}{|a_n|} = ...
0
votes
3answers
99 views

Is $ \int_0^1{\frac{\sin x}{x}dx} $ convergent?

I have this integral: $$ \int_0^1{\frac{\sin x}{x}dx} $$ And I should prove that it is convergent. I have understand that if the resulting area is finite, then this integral is convergent, right? ...