Tagged Questions
7
votes
1answer
101 views
convergence of series with $k!$
check if the following series converges:
$\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$
where $k!!=k(k-2)(k-4)(k-6)...$
I came across this exercise while going trough some old exams. I'm ...
0
votes
2answers
43 views
Checking $\displaystyle \int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$ for convergence
Given $\displaystyle\int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$, prove that it converges.
So of course, I said:
We have to calculate $\displaystyle \lim_{b \to \infty} ...
0
votes
2answers
20 views
Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$
Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac ...
3
votes
7answers
126 views
How to prove that $1/n!$ is less than $1/n^2$?
I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
0
votes
1answer
16 views
Absolute convergence.
Determine if absolutely convergent or not; Justify.
$$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$
if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
2
votes
3answers
28 views
behavior of a sequence
Imaging a sequence $ a_{k} \in \Omega $ with $ \Omega \subset \Bbb{R} $ closed, $ \lim\limits_{k \to \infty} \| a_{k+1} - a_{k} \| = 0 $.
My Professor said that because of this the sequence would ...
2
votes
2answers
33 views
What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$
What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$
I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
1
vote
1answer
49 views
Conditions for taking a limit into an infinite sum
Suppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: $$\lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to ...
2
votes
1answer
38 views
Radius of convergence of a power series with Bernoulli numbers
Say, we use the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
and then derive power series representations of the ...
0
votes
1answer
30 views
Calculate (& determine if converges): $\sum_{n=1}^\infty (-1)^n\frac 1n(0.5)^n$
Calculate (& determine if converges): $\sum_{n=1}^\infty (-1)^n\frac 1n(0.5)^n$
The above is a specific equation for $x_0=0.5$, but from here i'm pretty much stuck.
3
votes
1answer
67 views
Radius of convergence of: $\sum_{n=1}^\infty\frac {x^{n!}}{n!}$?
What the Radius of convergence of: $\sum\limits_{n=1}^\infty\frac {x^{n!}}{n!}$?
So far I tried finding the ...
2
votes
2answers
63 views
A question on the convergence of a Taylor series of some prominent function
The function $f:\mathbb{R}\to\mathbb{R}$ defined by
$$
f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\
0 & else \end{cases}
$$
is a prominent example of a function whose Taylor series ...
2
votes
3answers
91 views
How can I prove that $\int_1^\infty \left\lvert\frac{\sin x}{x}\right\rvert dx$ diverges?
I know a start could be to try and prove that $\int_1^\infty \frac{\sin^2x}{x} dx$ diverges since $\frac{\sin^2x}{x} \le \left\lvert\frac{\sin x}{x}\right\rvert$ in this interval, but I wouldn't know ...
2
votes
1answer
47 views
Radius of convergence of $\sum_{n=1}^{\infty} { (n \sin{\frac{1}{n}})^{n} x^n } $
We need to calculate the radius of convergence $R$ of:
$$\sum_{n=1}^{\infty} {\left(n \sin{\frac{1}{n}}\right)^{n} x^n }.$$
Here's what I did:
$$ \lim_{n\to\infty} { \left| ...
0
votes
3answers
78 views
Radius of convergence for $\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}} $
How do we find the radius of convergence $R$ of this power sum?
$$\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}} $$
How do we handle the $n!$ as the power of $x$?
1
vote
2answers
43 views
Finding the radius of convergence and what does it mean
We have just started learning this and I do not really understand it fully.
Given:
$$ \sum_{n=1}^{\infty} {(\pi^n + n + 1) \cdot x^n}$$
We should check what is the radius of convergence $R$ and what ...
2
votes
3answers
70 views
Show that $(x_n-y_n)$ converges to $x-y$.
Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$.
If it's asking about $(x_n+y_n)$. I know that I can ...
0
votes
1answer
29 views
Study the convergence of this sequence of functions
I have the following sequence of function:
$$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$
and I have to study its ...
1
vote
1answer
57 views
Series of Functions - Pointwise and Uniform Convergence.
I was hoping for some help for the following questions.
Prove that the series $\sum_{n=1}^\infty x^n(1-x)$ converges pointwise but not uniformly on $[0,1]$.
Prove that the series $\sum_{n=1}^\infty ...
1
vote
3answers
68 views
Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
I'm not familiar with proving divergent sequence. Do anyone have any des? ...
0
votes
1answer
51 views
Prove convergence of improper integral using change of variable.
This may be trivial, but I could use some help...
Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
1
vote
2answers
49 views
Prove or disprove a result for a double sequence.
Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
2
votes
2answers
88 views
What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?
How to solve the following limit question?
$$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$
Thanks a lot.
3
votes
1answer
62 views
When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?
Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$
Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$
Obviously, from the ...
9
votes
3answers
137 views
Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$
Evaluate the following limit:
$$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$
I have done the problem .
How I solved is
First I ...
2
votes
2answers
91 views
Interval of convergence of $\sum_{n=4}^\infty x^n/n^5$
Find the interval of convergence $$\sum_{n=4}^\infty x^n/n^5$$
I'm lost here. My intuition was to use the ratio test.
$$\lim_{n \to \infty} \frac{x^{n+1}}{(n+1)^5} \times \frac{n^5}{x^n} $$
...
2
votes
3answers
97 views
How to solve $\sum_{k=2}^\infty {\frac{1}{k^2-1}}$
I'm using the integral test to determine if this series converges. From what I have so far it seems that it diverges, but according to wolfram alpha it converges. Where is my mistake?
...
9
votes
5answers
135 views
Convergence of a sequence $c_n$
Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove ...
2
votes
1answer
195 views
binomial expansion formula proof, bases on Lagrange form of Taylor series remainder
Another exercise from Bartle/Sherbert Introduction to Real Analysis book (this one is exercise 9.4.14):
Use the Lagrange form of the remainder to justify the general Binomial Expansion
...
2
votes
0answers
36 views
Is this pointwise convergence sequence also uniform convergence?
$f_{n}$ and $f$ are continuous functions and $f_{n}\rightarrow f$ pointwise. Which of the following are correct?
$\int _{0}^{x}F_{n}\left( t\right) dt\rightarrow\int _{0}^{x}F\left( t\right) dt$
...
2
votes
1answer
48 views
convergence of complex series.
For what values of $(a,b\in \mathbb{C})$ does this series converge or diverge?
$\sum\frac{(k-a)^2}{(k-b)^3}$
if $a,b\in\mathbb{R}$ by the Limit comparison test (with $\sum\frac{1}{k}$ ) I know ...
2
votes
3answers
67 views
convergence of $\sum\frac{a_{n}}{n}$ if $\sum_{k=1}^{n}a_{k}\le M*n^{r}$ where $r<1$
Show that if the partial sums $s_{n}$ of the series $\sum_{k=1}^{\infty}a_{k}$ satisfy $|s_{n}|\le M*n^{r}$ for some $r<1$, then the series $\sum_{n=1}^{\infty}\frac{a_{n}}{n}$ converges.
1
vote
2answers
31 views
Radius of convergence in a series. Ratio test.
I am having a hard time with this question.
$$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$
I used the ratio test and got stuck here:
$$x^2 \lim_{k\to\infty} \frac ...
4
votes
1answer
106 views
Proof that $\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2}$ convergences uniformly using the Cauchy criterion
I'd like to use the Cauchy criterion to show that
$$f_n(x)=\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2} \mbox{convergences uniformly} $$
Here is what I did:
We want to show that $\forall \, ...
2
votes
3answers
93 views
Does this series converge or diverge?
I have a series here, and I'm supposed to determine whether it converges or diverges. I've tried the different tests, but I can't quite get the answer.
...
1
vote
1answer
52 views
Composition Taylor Series
Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name?
Thanks.
0
votes
1answer
31 views
Two quick questions about convergence (in the context of pointwise vs. uniform convergence)
I found this example online:
Let $\{f_{n}\}$
be the sequence of functions on $(0,\infty)$
defined by $f_{n}(x)=\frac{nx}{1+n^{2}x^{2}}$
.This function converges pointwise to zero.
...
5
votes
3answers
65 views
Representing Functions as Power Series
Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series.
Work thus far:
I separated it into two parts:
$$1/(1-x)^2 + x/(1-x)^2$$
I realize that the first expression is the derivative of $1/(1-x)$ and ...
1
vote
1answer
30 views
Specify conditions for $\alpha$ so that the iteration $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f.
Specify conditions on $\alpha$ so that the iterative process $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f if started with $x_0$ close to the root.
It is suggested that the proof should ...
2
votes
1answer
56 views
Value of limsup i?
This is a part of my question.
$\lim \sup \cos(n\pi/12)$ as n goes to infinity
What is the value of this limit?
3
votes
1answer
82 views
Cauchy but not fast cauchy
As the title indicates, I am trying to find a Cauchy sequence that is not fast (or rapidly) Cauchy. Could anyone suggest something?
A sequence $\{a_n\}_{n \in \Bbb N}$is termed fast (or rapidly) ...
5
votes
3answers
56 views
Convergence of definite integral
I have to find out the convergence of the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
Any help? Thanks
0
votes
2answers
29 views
What does $\sum_{n=0}^{\infty} (-1)^n .\frac{x^{2n+1}}{2n+1}$ converge to at x= 1 and x = -1
This is what I did.
$\sum_{n=0}^{\infty} (-1)^n .\frac{1^{2n+1}}{2n+1}=\sum_{n=0}^{\infty}(-1)^n .\frac{1}{2n+1}$
Now I broke it up to positive and negative.
$\sum_{n=0}^{\infty}(-1)^n ...
2
votes
1answer
90 views
A question on uniform convergence
Let $f,g:[0,1]\rightarrow \mathbb{R}$ be continuous functions. Define $$\displaystyle x_{n}(t)=f(t)+ \int_{0}^{t}x_{n-1}(s)ds\quad 0\leq t\leq 1,\;n=1,2,\cdots ,$$ where $x_{0}(t)=g(t),0\leq t\leq 1$ ...
2
votes
2answers
70 views
Checking convergence of an improper integral
I did a quick search here but couldn't find a similar problem (it's probably out there somewhere...)
I'm stuck with this rather simple improper integral:
$\int_{1}^{\infty} \frac{1}{x^{\alpha}-1}dx$ ...
3
votes
2answers
88 views
Question about convergence of series if $\{n a_n\} \to 0$
This is part of Rudin's PMA Exercise 3.14 (d). If I understand correctly, it would be helpful to prove the following:
Let $a_n$ be some sequence. Assume that $\lim_{n\to\infty} na_n = 0$.
Prove ...
1
vote
3answers
87 views
Uniform convergence of a sequence of functions
Let $(u_n(x))$ be a sequence of functions in $(0,\infty)$ such that:
$u_1(x)=x$, $u_{n+1}=\frac12\left(u_n(x)+\frac1{u_n(x)}\right)$ for $n\in\mathbb{N}$.
Check if $u_n(x)$ converge unfiormly in ...
1
vote
0answers
42 views
Convergence of integrals over divergent parts
I'm wondering if it is possible for an integral which diverges in the limits $1$ to $\infty$ to converge in the limits from $0$ to $\infty$. And if so: how could I find this out?
For example $$ ...
8
votes
2answers
122 views
$\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges? [duplicate]
Possible Duplicate:
Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent
If $\sum\limits_{n=1}^\infty |a_n|$ converges, the ...
1
vote
1answer
38 views
Convergence integral causal function
I have an exercise where there is the following given:
$f$ is a causal function.
$f$ is Laplace transformable:$\int_{0}^{\infty} f(t)e^{-zt}
\, dt = L(z) $ with $Real(z)> -1$
I have to ...
