Tagged Questions
0
votes
0answers
27 views
Are these series convergent?
I came across the following two series while trying to solve Laplace's equation in two dimensions.
$$T_1 = \sum_{n=1}^{\infty}\rho^n(A_n\cos(n\phi) + B_n \sin(n\phi))$$
$$T_2 = ...
2
votes
2answers
31 views
Find an interval of convergence and an explicit formula for $f(x)$
Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$
If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$.
The answers are $I = (-1,1)$ and ...
7
votes
1answer
99 views
convergence of series with $k!$
check if the following series converges:
$\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$
where $k!!=k(k-2)(k-4)(k-6)...$
I came across this exercise while going trough some old exams. I'm ...
2
votes
3answers
99 views
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$
Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n ...
2
votes
1answer
30 views
Intuition behind closed subsets of a metric space?
Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space.
Consider a metric space $$(X,d)$$
Then consider a subset of this space$$F$$
What the book ...
0
votes
0answers
40 views
funcitonal series convergence… SOS… [duplicate]
Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ?
i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
-1
votes
1answer
38 views
Alternating functional Series Convergence SOS…
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
1
vote
2answers
45 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
1
vote
1answer
29 views
series convergence
i ran into this question:
prove or show false:
if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is ...
7
votes
1answer
74 views
methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
2
votes
1answer
34 views
Convergence of sequence
Does the following:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
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 ...
1
vote
3answers
59 views
Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$
I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my ...
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
3
votes
3answers
51 views
Determine if a sequence converges using the number e
Knowing that the number $e =\lim_{n\to\infty}\left(1+{1\over{n}}\right)^n$ solve $a_n=\left({n+1}\over{n+3}\right)^n$
So (...)
$$\lim_{n\to\infty}\left({n+1}\over{n+3}\right)^n = ...
2
votes
1answer
67 views
convergence and divergence of a series
Let {$a_n$} be a sequence of non-negative real numbers such that the series $\sum^\infty_{n=1} {a_n}$ is convergent. If $p$ is a real number such that the series ...
0
votes
2answers
35 views
Series convergence with exponentials
I would like to understand if the following series converge (any closed form for that?!):
$$\sum_{n=0}^{\infty}\quad \frac{\exp(-n\cdot a)+n\cdot b}{(n+1)^2}$$
$$\sum_{n=0}^{\infty}\quad ...
1
vote
1answer
39 views
Sequence - Convergence?
I have to proof the following:
$ \lim\limits_ {n\to\infty} \dfrac{\sqrt[n]{n!}}{n} = \frac{1}{e}$
Do you have any hints for me, since I do not know where to start..
2
votes
2answers
51 views
Confusion regarding summing a matrix series
I have got following series while working out for an iteration problem.
$X_{k+1} = (1 + x + x^{2}+\cdots x^{2^{k+2}-3})$, $k = 0, 1, 2\cdots $ and $\|x\|<1$. My question is could I write $X_{k+1} ...
2
votes
1answer
54 views
Limit of a sequence in the space $\ell_2$
I have difficulties in the following problem.
Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that
$$
u^{k+1}=(1-\alpha)u^k+\alpha ...
9
votes
4answers
85 views
$a_{n+f(n)}-a_{n}\rightarrow0$ implies convergence?
$(a_{n})$ is a sequence of reals. Say $a_{n+f(n)}-a_{n}$ tends to 0 as n tends to infinity for every function f from the positive integers to the positive integers. Does this imply that $a_{n}$ is ...
2
votes
2answers
35 views
“Convoluted” sum convergence: $\sum_{i=0}^{n} a_i \sum_{k=0}^{n-i} b_k$
Suppose $\sum_{j=0}^{\infty} a_j =a$, $\sum_{j=0}^{\infty} b_j =b$. Is it true (or under what conditions is it true) that:
$$\lim_{n \rightarrow \infty}\sum_{i=0}^{n} \left( a_i \sum_{k=0}^{n-i} b_k ...
0
votes
2answers
40 views
Show uniform convergence of indefinite function series
How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5]
I thought about using the Weierstrass ...
1
vote
1answer
51 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 ...
1
vote
1answer
33 views
Numerical Analysis converging sequence question
Show that the sequence $p=10^{-2n}$ converges to zero with order $2$.
How many steps, $n$, will it take before this sequence is within $10^{-8}$ of zero?
Construct a sequence that converges with ...
2
votes
4answers
79 views
Radius of Convergence of power series of Complex Analysis
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
1
vote
2answers
39 views
Sequence convergence of positive numbers
Suppose that $\{a_j\}_j$ is a sequence of real numbers. Suppose for all $j$, $a_j \geq 0$ and the sequence $b_j = \frac{a_j}{1 + a_j}$ converges to $0$.
I wish to prove that $a_j$ converges to $0$.
...
1
vote
1answer
23 views
Convergence question of Dirichlet's Test
This is an after-chapter exercise of Dirichlet's Test.
Show that if the partial sum $S_n$ of the series $\displaystyle\sum_{k=1}^{\infty} a_k \leq Mn^r$, for some $r<1$, then the series ...
1
vote
1answer
70 views
Let $a_k=\dfrac{x^k}{k!}$ Show that $\dfrac{a_{k+1}}{a_k}\leq\dfrac12$
Fix $x\geq0$ and let $a_k=\dfrac{x^k}{k!}$
Show that $\dfrac{a_{k+1}}{a_k}\leq\dfrac12$ for sufficiently large k, say $k\geq N$
2
votes
2answers
83 views
Using the Banach Fixed Point Theorem to prove convergence of a sequence
Use the Banach fixed point theorem to show that
the following sequence converges. What is the limit of this
sequence?
$$\left(\frac{1}{3},
\frac{1}{3+\frac{1}{3}},
...
14
votes
5answers
286 views
Convergent or Divergent? $\sum_{n=1}^\infty\left(2^{\frac1{n}}-1\right)$
Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
...
0
votes
2answers
49 views
Ratio test and the radius of convergence
Let
$$
\sum_{n=0}^\infty c_n (z-a)^n
$$
be a power series. If the value
$$
r=\underset{n\to\infty}{\lim}\left|\frac{c_n}{c_{n+1}}\right|
$$
exists (the limit exists and is a real number), it is the ...
2
votes
1answer
47 views
Radius of convergence of $\sum_{n=1}^{\infty} { (n \sin{\frac{1}{n}})^{n} x^n } $
We need to calculate the radius of convergence $R$ of:
$$\sum_{n=1}^{\infty} {\left(n \sin{\frac{1}{n}}\right)^{n} x^n }.$$
Here's what I did:
$$ \lim_{n\to\infty} { \left| ...
2
votes
0answers
41 views
Bounding an implicitly defined sequence
I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined implicitly as
$$ \lambda_0 = \frac{1}{2},$$
and
$$\lambda_{k+1} = \max_{\lambda\in[1,b]} \left\{\frac{1}{2\lambda}\prod_{0\leq ...
2
votes
0answers
65 views
When $\ell^2$-convergence implies $\ell^1$- convergence?
Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$.
What are sufficient conditions on the sequence ...
2
votes
3answers
70 views
Show that $(x_n-y_n)$ converges to $x-y$.
Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$.
If it's asking about $(x_n+y_n)$. I know that I can ...
1
vote
2answers
37 views
Prove that a sequence diverges
Let $b > 1$. Prove that the sequence $\frac{b^n}{n}$ diverges to $\infty$
I know that I need to show that $\dfrac{b^n}{n} \geq M $, possibly by solving for $n$, but I am not sure how.
If I ...
1
vote
1answer
19 views
Question about convergence in $\mathcal D(\Bbb R)$
Let $\phi\in \mathcal D(\Bbb R)$. How to prove or disprove convergence of $\phi_n(x)=\frac{1}{n} \phi(nx)$ in $\mathcal D(\Bbb R)$?
I tried to do this by definition (we have to check two conditions ...
0
votes
1answer
48 views
How to prove a telescoping series converges ???
Let ${a_k}$ be a real sequence, such that $a_k \rightarrow 3, $as k$ $$\rightarrow \infty$ .
Prove that $\sum_{k=1}^\infty (a_k- a_{k+3}) = a_1 + a_2 +a_3 - 9$.
I know this is a telescoping series ...
4
votes
0answers
65 views
Divergence of the series $\sum\limits_{n=1}^\infty\frac{(n!)^n}{n^{4n}}$
After applying the root test to this series $$\sum_{n=1}^\infty\frac{n!^n}{n^{4n}},$$ I get $$\lim_{n\to +\infty} \frac{n!}{n^4}.$$ How would I complete this problem to show whether or not it is ...
1
vote
2answers
38 views
Convergence of $\max_{0\le i\le n}|f(i/n)|$
Suppose that $f\colon [0,1]\to\mathbb R$ is a continuous function. How can I prove that
$$\max_{0\le i\le n}\biggl|f\Bigl(\frac in\Bigr)\biggr|\to\sup_{0\le x\le1}|f(x)|$$
as $n\to\infty$?
Any help ...
1
vote
1answer
40 views
Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$
$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$
I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach ...
0
votes
2answers
40 views
Are these converging or diverging
I am having trouble working out the convergence of these series and was wondering if I could please have some assistance
a) $\displaystyle\sum_{n=0}^\infty\sin(e^n)\frac{n}{n^3+1}$
and
b) ...
4
votes
1answer
64 views
Is Cesaro convergence still weaker in measure?
I've encountered a question I couldn't answer, and I would appreciate any help:
Is it true that $f_n \xrightarrow{m}0$ $\Rightarrow$ $ \frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$?
Where ...
3
votes
4answers
100 views
How to prove that $\displaystyle \lim_{n \to \infty} \sqrt[n]{n}=1$? [duplicate]
I have seen this fact thrown around a lot and never really stopped to prove it; plugging in a few values convinced me of its truth. But I would like to see the result proved. To be clear, this is not ...
2
votes
2answers
71 views
Does the series always diverge?
Investigating the behavior of the following series:
$$\sum_{k=2}^\infty \frac{1}{\log^{p}k}$$
I broke it into 3 parts:
If $p = 0$ then it's just an infinite summation of ones, which diverges
If $p ...
0
votes
2answers
31 views
Investigate monotony, bound and convergence
I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it.
The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
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 ...
0
votes
3answers
41 views
finding values for absolute convergence
Find all values of real number p or which the series converges:
$$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$
I tried using the root test and the ratio test, but I got stuck on ...
