# Questions tagged [divergent-series]

Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

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### What is the proof of the theorem that the series of reciprocals of square-free numbers diverges?

I am not a mathematician. I am reading David Applebaum's book "Limits, limits everywhere ", in which he gives a proof of the divergence of the series of square-free numbers on page 91. I don't ...
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### When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A)$, etc., where $A$ is a matrix. Now my question ...
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### Investigate convergence of a series using comparison test

The series with $a_n$: $a_n = (n^{1/3}-(n-1)^{1/3})/n^{1/2}$ I tried comparing it to the $1/n^2$ and $1/n^{3/2}$ because those definitely converge, but proving the inequality gives rise to pretty ...
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### Determine the convergence of the series $\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$

Does $$\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$$ converge? I tried the ratio test but I failed.
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### Find the values of $\theta$ for which the series is convergent

How to find the values of $\theta$ for which the series $$\sum_{n=1} ^{\infty} \frac{(1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ ...+ \frac{1}{n})}{n} \cos n\theta$$ is convergent? What I could show ...
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### Show that $\sum_{n=1}^{\infty}\left(\frac{1}{(n+1)!}\prod_{k=1}^nf(k)\right)$ diverges.

Let $f:\mathbb{N}\setminus{\{0}\}\to\mathbb{N}\setminus{\{0}\}$ be an injective function. Show that $$\sum_{n=1}^{\infty}\left(\frac{1}{(n+1)!}\prod_{k=1}^nf(k)\right)$$ diverges. In the quotient ...
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### “Sum” of probabilities equals infinity

We have some independent trials with a certain chance of succeeding, say the $t$th trial succeeds with probability $p(t)$, and fails with probability $1-p(t)$. My question is, if the sum of ...
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### Geometric Series in Sigma Notation with natural log

So I am trying to solve for this problem below, I attach a screenshot of it, but I just can't seem to find the answer. Is anyone able to solve this? (Extra note: Un = Ur) So for (a) I got 2(1-2-10), ...
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### Convergence of this series: $\sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{\sqrt{2n^2+1}}(x-1)^{2n}$? [closed]

I have been trying for a long time to find an approach to this problem! I can't figure out how I could determine the convergence of this series!! I tried to determine the radius of convergence with ...
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### Convergence of the Average of Partial Sums

The Maclaurin series of $\frac{1}{x+1}$ is of course $\sum_{n=0}^{\infty}\left(-1\right)^{n}\left(x\right)^{n}$, and its interval of convergence is only considered to be $-1<x<1$. Though, it ...
Test series for convergence / divergence using a comparison test: $$\sum_{n=1}^\infty\frac{n^2+1}{n^3+2}$$ Now, If it would be $$\sum_{n=1}^\infty\frac{n^2+1}{n^3-2}$$ then I could compare it as ...