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
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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
122
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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 \...
- 17.5k
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
$$
S_n := \sum_{i=0}^{n-1} \frac{t_{i+1} - t_i}...
- 17.5k
-1
votes
1
answer
88
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Tighter upper bound of $\sum\limits_{k=1}^\infty\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$
How do I find tight upper bounds of $$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 ...
- 6,539
2
votes
2
answers
166
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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:
$$\sum_{n=1}^{\infty} (-1)^n\ln \left [ 1+\sin \left (\frac{\pi}{n\sqrt{n}} \right) \...
- 67
3
votes
2
answers
123
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Convergence of $\sum_{n=1}^{\infty} \sqrt{\frac{\ln(n+1)}{1+\ln(n+1)}}\left(\sqrt{\frac{1+\ln(n+1)}{1+\ln\left(n\right)}}-1\right).$
I came across the sum $\sum_{n=1}^{\infty} a_n$ while computing an upper bound, where $$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\...
- 3,290
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
295
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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
82
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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
559
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:
$$n!^{2/n}...
- 6,722
3
votes
1
answer
144
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Does series converge absolutely or conditionally?
I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally.
$$\sum^{∞}_{k=1} \frac{\sin(2k^2+1)}{k^{3/2}}$$
I ...
- 143