# Tagged Questions

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### Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...
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### Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? Is my solution correct?

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? I am confused because my friend insists the series converges conditionally. I think the series diverges. Here is my process and solution: ...
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### $\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
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### Prove or disprove the convergence of…

I need help with the following problem, please help. For positive real x. Let $${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$ Prove or disprove the convergence ...
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### Convergence of $\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$

$\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$ A) For which $p\in \mathbb{R}$ is the series convergent? B) For which $p\in \mathbb{R}$ is the series divergent, and what is ...
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### Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
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### Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
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### Does it converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get 1. I suppose I should compare it with some other series, but I can't figure out with ...
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### Determine whether the following series convergent? [closed]

Is the following series convergent? $$\sum_{k = 1}^{\infty}{1 \over k\left[1 + \ln^{2}\left(k\right)\right]}$$
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### Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}.$$
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### The series $a_n$ is conditionally convergent. Prove that the series $n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $|a_n|$ is not necessarily smaller than $n^2a_n$. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0$ ...
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### Determining convergence/divergence

I am studying for my exam tomorrow and have come across some problems I cannot get. I have put them below with what I have tried/thinking process behind. Thank you. ...
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### Does the following series converge or diverge?

(2n-1)/(n!) I used the ratio test here and got the lim as n -> infinity to be 0. Therefore, I assumed that the series converges. However, my textbook says that it ...
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### Showing how this infinite sum diverges [duplicate]

$\displaystyle{\sum_{n=1}^{\infty} \left[\left( 1 + {1 \over n}\right)^{n} - {\rm e}\right]}$ I tried both root and ratio tests (for the root test, the expression became way too complex to handle) ...
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### Does the Leibniz Test say that if the sequence doesn't go to 0 that the series diverges?

My book explains the Leibniz test by saying: "Assume a sub n is a positive sequence that converges to 0..." And goes on to say that that means the alternating series converges. What if the sequence ...
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### Find values of $p$ for which the series is convergent.

I am still having a hard time understanding how to find what values of $p$ allow for the series to converge. Here is the series I am currently trying to find $p$ for: $$\sum_{n=1}^\infty n(1+n^2)^p$$ ...
### Find the values of p for which $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent
Find the values of p for which the series $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent. I know that the p-series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent if $p>1$ and divergent if ...
### Is the series $\sum_{n=1}^\infty \frac{1}{n^3+n}$ convergent or divergent? [closed]
Is the series $\sum_{n=1}^\infty \frac{1}{n^3+n}$ convergent or divergent? And how? What method is required? Thanks.