# Tagged Questions

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### Limit of infinite sum

What is the answer to the following limit? $$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
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### Computation of standard series

I am stuck on the computation of the following sum: $\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
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### Finding the limit $\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$

Find the following limit: $$\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$$ for all $\lambda>0$.
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### Prove that for every $p>0$, $\lim_{n\rightarrow∞}\int_n^{n+p}{\sin x\over x} = 0$

Got stuck with this question: Prove that for every $p>0$, $\displaystyle \lim \limits_{n\rightarrow∞}\int_n^{n+p}{\sin (x)\over x} = 0$. Thanks in advance for any help!
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### Investigate monotony, bound and convergence

I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it. The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
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### Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.

Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition. I'm not familiar with proving divergent sequence. Do anyone have any des? ...
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### $x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
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### Finding the infinite product $\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right)$

How do I find: $$\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right) \quad$$ I am pretty sure that the infinite product converges, but if it doesn't please let me know if I have made an error. ...
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### Find the limit of the sequence $\frac{n!}{2^n}$ as n tends to infinity

We've only been taught to find limits using the Squeeze Theorem and L'Hopitals Rule, so I'm not sure how to go about finding the limit of this sequence.
### Find the supremum of $\left ( n+1 \right )^{\frac{2}{n^2}}$
As in the topic, my task is to find supremum and infimum of a given set $$f(n):=\left ( n+1 \right )^{\frac{2}{n^2}}, n\in \mathbb{N}$$What is funny, I managed to do this task few weeks ago and I ...