# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Does this series converge?

I showed that, by a combination of the root test and Stirling's approximation, the series $$\sum \frac{n^n}{n!}$$ converges (the ratio test is inconclusive.) However a solution that I am comparing my ...
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### Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
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### Number sequence [closed]

The following terms are found by alternately adding 4 and 6 to the previous term. The first six terms are 13, 17, 23, 27, 33, 37. (a) Find the 80th term. (b) The nth term is 203. Find n. Note: ...
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### $a_{3n} = {-1}/ \sqrt[3]n, \ a_{3n+1} = {-2}/ \sqrt[3]n, \ a_{3n+2}= {3} / \sqrt[3]n$ then $\sum_{n=1}^{\infty} a_n$ converges

Let $a_n$ defined by: $$a_{3n} = \frac{-1}{\sqrt[3]n},\quad a_{3n+1} = \frac{-2}{\sqrt[3]n},\quad a_{3n+2}=\frac{3}{\sqrt[3]n}$$ show that $\sum_{n=1}^{\infty} a_n$ converges. I thought about ...
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### proof of $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$

How to prove $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$? $\ \ \ \ \sum_i a_i = 1$, $0\leq a_i \leq 1$ and $y,x_i\in \mathbb{R}^m, \ \ \forall i$ It seems simple however I ...
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### The growth of a cyclic sequence with $n$ terms under the rule $x_i \mapsto x_{i+1}+x_i$ and $x_{n}\mapsto x_1+x_n$

Say we have $100$ terms connected in a circle, starting with $x_1$ and going clockwise to $x_2$, to $x_3$, etc. and with $x_{100}$ going to $x_1$. Now initially $x_1 = 1$ and $x_2=-1$ and the rest of ...
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### Pointwise convergence of a sequence of functions $g_n$

Let $$g_n(x) = \begin{cases} 1 & \text{if } x = \frac{1}{n}\\ x & \text{if }x = 1, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n-1}\\ 0 & \text{otherwise} \end{cases}$$ I am trying to figure ...
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### Convergence of the series

Find the convergence and absolute convergence of the series $∑\frac{(-1)^{n+1} n}{1+n^2}$ For Absolute convergence, I found out by comparison tests, it is not absolutely convergent. But I couldnt ...
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### Finding recurrence relation 2 - Ternary sequences. [duplicate]

for $n \ge 1$ , Let $f(n)$ be the number of ternary sequences {$0,1,2$} of length n, that the sum of their characters are even, and they have at least one show of $0$. How do i find the recurrence ...
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### Proof verification: another convergent sequence proof

Note: Sorry, I posted this earlier with a glaringly obvious error - here's the improved version: The statement I'm trying to prove is: Let $(x_n)$ be a convergent sequence and $K \in \Bbb N$. ...
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### What is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ equal to?

So we know that $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} = 1$$ Is there any information on what $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^{n^2}}$$ equals?
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### Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$

Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$. I have already proved that a function is Riemann integrable if and only if it is bounded and continuous a.e. If $f$ is bounded, ...
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### Limit of the sequence $n(e^{1/n}-1)$ as $n \to \infty$

How can I evaluate $$\lim_{n\to \infty} n\left(e^{1/n} - 1\right)$$ I have tried it applying the Squeeze Theorem but I can't seem to get anywhere with that.