# Tagged Questions

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### Convergence of sequences such as $B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$

Examine the following arithmetic sequences if they converge or do not.The first one is $$B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$$ and the second $$C(n)=n/(n^2+1)+\dots+n/(n^2+n)$$ It was on our ...
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### Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute $$\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$$ According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
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### Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}}$$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
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### Convergence or divergence of the integral $\int_0^1 dx/\sin x$

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx$$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
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### Does it converges? $\sum \frac{\sqrt{n}\ln(n)}{n^2+1}$

As I checked on Wolfram Alpha I know that $$\sum \frac{\sqrt{n}\ln(n)}{n^2+1}$$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
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### Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
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### How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
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### Convergence of function

Suppose that $V(t)$ is nonnegative continuous function ($\forall t:V(t)\ge0$). $\dot{V}(t) = -|h(t)|^2 + f(t)g(t)$ $f(t)$ is a bounded and uniformly continuous function. $g(t)$ is a bounded and ...
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### How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
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### Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
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### Decide convergence divergence of $\sum \dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Does the series $\sum \dfrac{1}{(\ln n)^{\ln n}}$ converges? I can intuitively say that it converges, because $(\ln n)^{\ln n}$ is going to $\infty$ on a hayabusa
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### Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
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### Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
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### Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
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### Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
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### How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
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### Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$\sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots$$ ...