# Tagged Questions

Convergence of sequences and different modes of convergence.

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### Is Taylor's series for cos uniformly convergent?

Is it true that Taylor's series for cos uniformly convergent (for all $\mathbb{R}$)?
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### Monotonicity, boundaries and convergence of the sequence $\left\{ \frac{a^n}{n!} \right\}$.

everyone. I have a doubt on the following question: Let $\left\{ \frac{a^n}{n!} \right\}, n \in \mathbb{N}$ be a sequence of real numbers, where $a$ is a positive real number. a) For what ...
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### Proof convergency of series $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!}$

I have used Cauchy and came to step where i have $\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \ldots + \frac{1}{(n+p)!}$ i cant find upper boundary $\epsilon$ , hope you guys can help me. Thanks in ...
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### convergence of sequence with factorials

i want to study wether the sequence $$u_n=\frac{1}{n!}\sum_{k=1}^nk!$$ converges or not. I didn't get the trick yet, do you have any clue?
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### Proving a partial sum is convergent

Assume $a_n$>0 and lim($n^2a_n$) exists. I need to show that the sequence of partial sums of ($n^2a_n$) converges. I know that ($n^2a_n$) is bounded and increasing. I can also show that its sequence ...
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### Showing partial sums converge based on sequence convergence

Assuming $a_n$$>0 and that lim(n^2a_n) exists, I need to show that \sum$$a_n$ converges. My plan is to show that the sequence of partial sums, let's say $s_n$ of ($n^2a_n$) converges, and ...
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### If $(u_n)$ is bounded and $\lim u_n^2+u_n-u_{n+1}=0$ then $u_n \to 0$

Let $(u_n)$ be a real bounded sequence. Suppose that $\lim\limits_{n \to \infty} (u_n^2+u_n-u_{n+1})=0$. Prove that $u_n \to 0$. I was able to develop a prove, looking at the map $x \mapsto x^2+x$ ...
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### When is it ok to use a sequential limit in place of a continuous limit?

I am working through some Lebesgue integral problems, and I've come across a few instances where I would like to use the dominated/monotone convergence theorems, but the limit is continuous, and I'm ...
I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...