All Questions
Tagged with sequences-and-series divergent-series
1,166
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An example of infinite divergent series giving rational fraction of Pi.
Can an example of divergent integer sequence along some regularization method be found where the generalized sum is $c π^k $, k integer and c being a rational?
I only know π appearing, but not as ...
- 9
2
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2
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179
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Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
- 825
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Alternating series comparison test
Let's say I have two alternating series of terms,
$(-1)^n A_n$
$(-1)^n B_n$
If I know (by for example Leibniz criteria) that one of the series converges / diverges, can I use comparison criteria to ...
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3
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Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$
I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$.
In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
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Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
- 1,145
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Show that $\sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}$ with $ \epsilon > 0$ diverges [duplicate]
I think this sum diverges but I can't seem to show it.
$$ \sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}, \ \ \ \epsilon > 0 $$
I have tried using the bound $n \log n = \log n^n < n^n$ which led me ...
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Show that for $(u_{n})$ strictly decreasing sequence tending to 0, $\sum_{n=0}^{+\infty} \frac{u_{n}-u_{n+1}}{u_{n+1}}$ diverges
I come to ask you about a problem coming from a serie's exercice sheet brought by a student that I can't crack.
Let $(u_{n})$ be a strictly positive decreasing sequence which converges to 0.
How do I ...
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How to prove Exercise 8.2.6 from Analysis 1 Terence Tao [duplicate]
I have been stuck a while on the following exercise of Analysis 1 from Terence Tao:
Exercise 8.2.6 Let $\sum^\infty_{n=0}a_n$ be a series which is conditionally convergent, but not absolutely ...
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How to find the divergent renormalization of $\sum_{n=1}^{\infty} \frac{2^n}{n}$?
Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n} $$
This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n} $ and 'evaluating' that power ...
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How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary?
I'm looking for an example of a series of complex functions $\{f_n(z)\}$ with the following properties:
The series $\sum_{n=1}^{\infty} f_n(z)$ converges uniformly on a closed region $D$ in the ...
- 15
2
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1
answer
115
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Limit $\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$ [duplicate]
I am currently trying to calculate the following limit of sequence: $$\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$$ I need it to prove that a series diverges, but I ...
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How does one show that an alternating series $\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$ is unbounded/diverges? [duplicate]
I have arrived at the point where I have a series
$$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$$
that I know should diverge (checking via python implies it might be divergent, and wolframalpha times out ...
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53
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Convergence of a finite series [closed]
How do I write a converged form for the following finite series?
$\sum_{k=0}^{M-1}\frac{1}{k!}f(x)^k$ where M is any integer and x is variable
If the convergence of this series is not possible, is it ...
- 19
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Does this sequence of real number converge?
Given the sequence of real numbers $$a_n = \left(\sum_{k=1}^n \frac{1}{\sqrt{1+k^2}}\right)- \ln(n+\sqrt{1+n^2}) , n\in \Bbb{N}$$
Test its convergence.
What I did was to separate the sequence from the ...
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Proof of non-convergence of sine series
2 By considering the identity $\cos[(2n-1)\alpha]-\cos[(2n+1)\alpha]\equiv2\sin\alpha\sin2n\alpha,$ show that if $\alpha$ is not an integer multiple of $\pi$ then $\sum_{n=1}^N \sin (2n\alpha) = \...
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