All Questions
104
questions
1
vote
1
answer
123
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Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
0
votes
0
answers
104
views
Is this divergent series, convergent?
Examining the series $\sum_{n=1}^{\infty} \frac{1}{nx}$ alongside its integral counterpart reveals insights into its convergence. Notably, the integral over intervals from $10^n$ to $10^{n+1}$ yields ...
- 81
1
vote
1
answer
123
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Show that $\left | \sum_{k=1}^\infty \frac{\left(e^{ik} + \sin(\sqrt 2 k) + \cos(\sqrt 3 k)\right)^3}{k} \right| < 6$
I am trying to show the following sum is bounded:
$$ \sum_{k=1}^\infty \frac{\left(e^{ik} + \sin(\sqrt 2 k) + \cos(\sqrt 3 k)\right)^3}{k}$$
and to show that the magnitude
$$\left | \sum_{k=1}^\infty \...
- 972
1
vote
1
answer
40
views
Is $\alpha :=\sup_{n \ge 2} \sup_{1 \le k \le n} S_{n, k}$ finite or not?
Given a natural number $n \ge 2$, we define
a finite sequence $(t_{n,i})_{i=0}^n$ by $t_{n, 0} := 0, t_{n, n} := \frac{\pi^2}{6}$ and
$$
t_{n, i+1} := t_{n, i} + \frac{1}{ (n - i)^2}
\quad \forall i \...
- 17.6k
1
vote
1
answer
48
views
Is there a sequence $(t_n)_{n\ge 0}$ such that $\lim_n t_n < \infty$ and $\sup_n S_n < \infty$?
Let $(t_n)_{n\ge 0}$ be a strictly increasing sequence of non-negative real numbers such that $t_0 =0$. We define an induced sequence $(S_n)_{n\ge 1}$ by
$$
S_n := \sum_{i=0}^{n-1} \frac{t_{i+1} - t_i}...
- 17.6k
-1
votes
1
answer
88
views
Tighter upper bound of $\sum\limits_{k=1}^\infty\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$
How do I find tight upper bounds of $$S=\sum_{k=1}^{\infty}\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$$ ? The derivative of $\tanh$ is $\text{sech}^2$, so using $$S=\sum_{k=1}^\infty\frac{1}{\cosh^2(\cosh ...
- 6,549
2
votes
2
answers
169
views
Proving Absolute Convergence of $\sum_{n=1}^{\infty} (-1)^n\ln \left [ 1+\sin \left (\frac{\pi}{n\sqrt{n}} \right) \right ]$
I am trying with no success to prove the Absolute/Conditional Convergence / Divergence of the following series:
$$\sum_{n=1}^{\infty} (-1)^n\ln \left [ 1+\sin \left (\frac{\pi}{n\sqrt{n}} \right) \...
- 67
3
votes
2
answers
123
views
Convergence of $\sum_{n=1}^{\infty} \sqrt{\frac{\ln(n+1)}{1+\ln(n+1)}}\left(\sqrt{\frac{1+\ln(n+1)}{1+\ln\left(n\right)}}-1\right).$
I came across the sum $\sum_{n=1}^{\infty} a_n$ while computing an upper bound, where $$a_n = \sqrt{\frac{\ln\left(n+1\right)}{1+\ln\left(n+1\right)}}\left(\sqrt{\frac{1+\ln\left(n+1\right)}{1+\ln\...
- 3,360
0
votes
0
answers
30
views
Help needed with finding power series, convergence radius, and interval
I am currently struggling with finding the power series, convergence radius, and interval for the following functions:
a) $f(x)=\frac{2}{1-x}$
b) $f(x)=2 \ln (1-x)$
Here is what I have attempted so ...
- 39
1
vote
1
answer
297
views
Proof that $\sum_{n=2}^{\infty}\frac{1}{n\sqrt{\ln{n}}}$ diverges? [duplicate]
Probably the best way would be Limit Comparison Test. Intuitively by looking at $a_n$ the sum probably diverges.
However, let's say I choose a $b_n$ in the form $\frac{1}{n^p}$ where $p>1$, which ...
- 467
1
vote
2
answers
83
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Determine the end behavior of $\sum_{n=0}^{\infty}\frac{2^n}{3^n+4^n}$ using Root Test
I took the nth root of $\frac{2^n}{3^n+4^n}$, and while the numerator simplifies nicely to 2, the denominator is totally stumping me. Since this would all be inside of an infinite limit, we must ...
- 467
1
vote
2
answers
83
views
Finding the sum of the series $\sum_{n=2}^{\infty} ((n^2+1)^{1/2} - (n^3+1)^{1/3})$
I need help in finding the sum of the series $\sum_{n=2}^{\infty} ((n^2+1)^{1/2} - (n^3+1)^{1/3})$ if it converges.
I can't even prove convergence. I tried comparison test. I tried telescoping or even ...
- 73
2
votes
4
answers
568
views
prove that $\ln(2) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}$ without using Abel's theorem [duplicate]
I know that
$$\ln(1+x) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}x^k$$
for $|x| < 1$. But how can I show
$$\ln(2) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}$$
without using Abel's theorem? There's a ...
- 1,225
2
votes
1
answer
70
views
Convergence of $\sum_{n=1}^\infty 1/(n!)^{2/n}$
I'm trying to investigate the convergence of $\sum_{n=1}^\infty \frac{1}{(n!)^{2/n}}$
I know that $n!\geq (n/2)^{n/2+1}$ for all $n\geq 1$, and so after taking $2/n$ to both sides, I obtain:
$$n!^{2/n}...
- 6,722
3
votes
1
answer
144
views
Does series converge absolutely or conditionally?
I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally.
$$\sum^{∞}_{k=1} \frac{\sin(2k^2+1)}{k^{3/2}}$$
I ...
- 143
-3
votes
2
answers
72
views
Find if Series Converges or Diverges Question. [closed]
I was wondering if anyone knows what is the best test to use to find if this series converges or diverges.
$$\sum^{∞}_{k=1} \frac{4k^5}{3^k+4k^3}$$
Edit: Sorry for the bad question, This was my first ...
- 143
3
votes
2
answers
221
views
Convergence of $\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I need help with this.
$\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I know that it converges but i can not proove why.
I tried to rewrite it, it seems to be a geometric serie. I tried to do a common ...
- 47
0
votes
0
answers
41
views
An infinite summation involving spherically symmetric eigenfunctions of the heat equation
I need help solving for the $C_n$ in the following infinite sum:
$$−\sin(\pi r) = \sum_{n=0}^{\infty}C_n\frac{\sin(n \pi r)}{r}\cdot e^{-n^{2}\pi^{2} t}$$
I have been trying to solve this for a while ...
- 15
-1
votes
1
answer
81
views
Converges or not? $ \sum_{n=1}^{\infty} (-1)^n \frac{2n+1}{n} $
$$ \sum_{n=1}^{\infty} (-1)^n \frac{2n+1}{n} $$
An answer sheet says that this is a divergent series but my computation says that it is convergent. What could be my error?
My solution:
$$ \sum_{n=1}^{\...
user912953
4
votes
1
answer
690
views
Formula for nth derivative of partial sum of geometric series.
I am trying to find a formula for either
(1) the $n$th derivative for the following $m$th partial sum:
$$\frac{d^n}{dx^n} \sum_{i=0}^m x^i$$
or (2) the $n$th derivative of the infinite series given by
...
- 6,099
2
votes
3
answers
103
views
$\sum_{n=1}^{\infty} u_n^2=0$ $\Rightarrow$ $u_n=0 \ \forall \ n\in \mathbb{N}$
Let, $<u_n>$ be a real sequence and given that $$\sum_{n=1}^{\infty} u_n^2=0$$
Prove that $u_n=0 \ \forall \ n\in \mathbb{N}$
Attempt
$u_n^2\geq 0 \ \forall \ n\geq 1$
Since, $$\sum_{n=1}^{\...
user908035
1
vote
1
answer
133
views
Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $
Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $ where $f(x)= sin(log_e x)(\frac{1}{x^{a}}-\frac{1}{x^{1-a}})$,$ 0<a<1/2$
My try -
$\lim_{N\to +\infty}[ \int_{1}...
user826941
3
votes
2
answers
154
views
Convergence of $S_p = \sum_{k=1}^{p}\left( \sum_{j=1}^{k}\frac{r^{k-j}}{j}\right)^2$
I'm looking at the following sum
$$ S_p = \sum_{k=1}^{p}\left( \sum_{j=1}^{k}\frac{r^{k-j}}{j}\right)^2$$
where $r \in (0,1), p > 1$. The goal is to inspect whether the sum converges to some ...
- 740
0
votes
1
answer
131
views
Divergence of $\sum_{n=1}^{\infty}\frac{a \cos(b \ln n)+b \sin(b \ln n)}{n^a}$
Divergence of $$\sum_{n=1}^{\infty} \frac{a \cos(b \ln n) + b\sin(b \ln n)}{n^a}$$ where $0<a<1, b>0$ are constants.
My try:
$$\cos(x)\geq 1-\frac{x^2}{2} , \forall x$$
$$\sin (x)\geq -x , \ ...
user846558
0
votes
1
answer
49
views
How to solve an integral inside a summation (with a divergent term)
I have a question that might be silly, but I really don't understand what is going on. I have to solve the following integral:
$$ \sum_{n \in \mathbb{Z}} \int_{m}^{m+1}e^{inx}dx $$
However if we try ...
- 91
0
votes
1
answer
61
views
Alternating geometric series, not sure what i am looking at
I have written out a series describing a model system but i cannot find if there is a historical representation of the like series:
$$N_s= N\sum_{i=1}^n(-1)^{i-1}P^i$$
It appears to be an alternating ...
1
vote
1
answer
54
views
Check series for convergence or divergence
$$\sum_{n=1}^{\infty} \frac{2+(-3)^n}{4^n} $$
I had an idea to decompose this series into the sum of two series
$$ \sum_{n=1}^{\infty} \frac{2+(-3)^n}{4^n} = \sum_{n=1}^{\infty} \frac{2}{4^n} + \sum_{...
- 13
3
votes
1
answer
137
views
Converging / Diverging sum with a constant power:
I need to prove this sum is diverges/convergent/conditional convergent , but I am pretty sure it is converging to a value:
$$\sum_{k=1}^{\infty} \frac{(1+\frac{1}{k})^{k^a}}{k!}$$
For some constant:
$...
0
votes
2
answers
57
views
Prove $ \sum_{n=1}^\infty \frac{(-1)^n }{2n\cdot4^n} = \ln(2/5^{0.5})$ [closed]
I proved that the following summation converges but how to prove that the it is equal to $\ln(2/5^{0.5})$
$$ \sum_{n=1}^\infty \frac{(-1)^n }{2n\cdot4^n}$$
- 21
0
votes
2
answers
57
views
Finding radius of convergence.
I was asked to prove that $f(x)$ is derivative in $x=1/2$ but I found that the radious of convergence is $0$, what did I do wrong?
$$ f(x) = \sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{2n(2n+1)}$$
I put ...
- 21