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### Convergence of the sequence $x_{n} = \int_{1}^{n}\frac{\cos t}{t^{2}}$ as n tends to infinity.

$|x_n-x_m| \le \left|\int_n^{m} \frac 1 {t^{2}}dt\right| \to 0$ so $(x_n)$ is Cauchy.
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### Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$

Just use Integral test, We have $$\sum_{n \geq 1} \frac{\ln(n)}{n}$$ We know that it converges iff integral $$\int_1^\infty \frac{\ln(x)}{x} \, \mathrm{d}x$$ converges. But, \begin{align*} I &= \...

### Finding a sequence for two different series so that both converge

Take $a_n=(-1)^{n}n^{-1/2}$ the first series converges by Alternating Series Test and the second one converges absolutely.
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1 vote
Accepted

• 2,380
Accepted

• 6,783
Accepted

### Check convergence of a trigonometric series

That's a series of numbers greater than $0$, and so the comparison test can be used. You have\begin{align}\lim_{n\to\infty}\dfrac{\sin\left(\dfrac1{\sqrt n}\right)\tan\left(\dfrac1{\sqrt n}\right)}{\...

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