# 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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### Evaluating the limit of a sequence using Squeeze Theorem

Let $a \in \mathbb R$, $0 < a < 1$. Find $$\lim_{n\to\infty}\left(\frac{a^n+a^{2n}}{1+a^3}\right)^\frac1n$$ I am supposed to use the Squeeze Theorem, so I tried the following, but I don't ...
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### Using Partial Summation to evaluate a series

$$S = \sum_{x=1}^{\infty} \frac{\sin(x)}{x}$$ Using partial summation. Obviously, $$S = \lim_{n \to \infty} \sum_{x=1}^{n} \frac{\sin(x)}{x}$$ Partial Summation: \begin{align*} \sum_{n=1}^{N} a(n)...
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### Subsequences and upper and lower limits of a sequence

I'm working on a homework assignment in which I have to find the upper and lower limits of a sequence. I've partitioned the sequence into two subsequences (one consisting of all even terms and ...
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### I need help using the limit comparison test for $\sum \frac{1}{\sqrt{n^2 + 1}}$

I need to determine whether the following series converges or diverges: $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^2 + 1}}$$ I'm having trouble finding a series to compare this to but I was thinking ...
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### Closed form for $\frac{H_k}{k^2}$.

I was trying to solve some problem and came across the following series: $$\sum_{k=2}^{\infty}\frac{H_k}{k^2}$$ I tried to find a closed form for that series but could not. Also I looked some ...
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### Find value of limit $a_n=(1+2^n+3^n+…+n^n)^{1/n}\sin(1/n)$

I want to find the limit $\lim_{n\rightarrow\infty}(1+2^n+3^n+...+n^n)^{1/n}\sin(1/n)$. I have tried so far to bound it but I hadn't had any success.
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### example of a sequence which is equivalent to zero at infinity but does not converge to zero

I want to know if there is an example of sequence $(u_n)$ which is equivalent to zero at infinity but does not converge to zero? $$\exists ? (u_n) \;|\; u_n \sim 0 \mbox{ but }u_n\nrightarrow 0.$$ ...
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### General form of $\sum_{i=1}^{k} \frac{1}{k}\tan(\frac{i\theta}{k})$

I have tried calculating this equation by setting a large value of $k$. given $\theta$, $$S = \sum_{i=1}^{k} \frac{1}{k}\tan\bigg(\frac{i\theta}{k}\bigg)$$ When increasing $k$, $S$ seems converging ...
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### A question regarding a small calculation in the proof of a theorem.

I understand the theorem overall, and that the $\epsilon$ values are arbitrary. Notice how the values $\epsilon_1=\frac\epsilon{2M}$ and $\epsilon_2=\frac\epsilon{2(|t| + 1)}$ are just assumed, and ...
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### Baby Rudin Chapter 3 Problem 11(d)

Suppose that $a_n > 0$ for all $n \in \mathbb{N}$ and that $\sum_{n=1}^\infty a_n = +\infty$. Let $b_n \colon= {a_n \over {1+na_n}}$ for all $n \in \mathbb{N}$. Then we can show the following ...
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### What's the convergence of this summation?

$$\sum_{i=1}^{n}a^{i-1}=a^0+a^1+a^2+...+a^{n-1}=1+a+a^2+...+a^{n-1}$$ Looks like a geometric series. Is there a more compact formula for the convergence of this summation? Thanks for any help you ...
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Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: $$\sin(x)=x\prod_{i=1}^{\infty}(1-\frac{x^2}{i^... 4answers 109 views ### Why is \lim_{n\to\infty} n(e - (1+\frac{1}{n})^n) = \frac{e}{2} I'm having trouble understanding why$$\lim_{n\to\infty} n(e - (1+\tfrac{1}{n})^n) = \frac{e}{2}$$Can someone offer me a proof for this? 2answers 90 views ### Euler sequence, limit of an related sequence Study the convergence of the sequence  \left( a_n\right)_{n\in\mathbb{N^*} } defined by$$ a_{n}+e^{a_{n}}=\left( 1+\frac{1}{n}\right) ^{n}+1,~\forall n\in \mathbb{N} ^{\ast} $$and find its ... 0answers 87 views ### Proving not equicontinuity in \Bbb R but equicontinuity in any other closed subset of \Bbb R Let F = {f_{n} | n ∈\Bbb N } be an infinite collection of functions f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R. Prove that F is not equicontinuous on \Bbb R but equicontinuous on [−a, a] for any a ... 3answers 210 views ### Can the inequality \sum\limits_{i=1}^n\frac{1}{\sqrt{i}} < 2\sqrt{n} - 1 be proved without induction? Maybe some of you have seen one of the posts where the inequality \sum\limits_{i=1}^n\frac{1}{\sqrt{i}} < 2\sqrt{n} is proved by induction (here and here). It can be proved without induction too, ... 1answer 86 views ### Clarification for an Example in Rudin This examples (example 7.21, page 156) proves that the given sequence of functions has no uniformly convergent subsequence: Let$$f_{n}(x)=\frac{x^2}{x^2+(1-nx)^2},\quad (0\leq x\leq 1, n=1,2,3,...
I want to find a closed form for $$\sum_{i=0,1,...}{\left\lfloor\frac{n}{2^i}\right\rfloor}$$ e.g. ...
### Series expansion of $\exp(-x)$ using powers of $\frac{1}{1+x}$?
I want a power series expansion of $e^{-x}$, but since powers of x blow up as x→∞ and powers of $\frac{1}{x}$ blow up as x→0, I was wondering if a series expansion of $e^{-x}$ using powers of \$\frac{1}...