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

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1answer
67 views

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 ...
0
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1answer
31 views

Prove convegent sequences space has the same dimensionality as bounded sequences and series converging to $0$ spaces.

Let $K$ be a field ($\mathbb{R}$ or $\mathbb{C}$), and let $$c(K)=\{x:\mathbb{N}\mapsto K : \lim{x_n}\text{ exists}\},$$ be the vector space of converging sequences and $$c_0(K)=\{x:\mathbb{N}\mapsto ...
1
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1answer
86 views

Dirac's delta, infinite series and integral

Why $\int_{-\infty}^x\sum_{i=1}^{+\infty} p^{i-1}\delta(\alpha-i)d\alpha = \sum_{i=1}^{+\infty}\int_{-\infty}^x p^{i-1}\delta(\alpha-i)d\alpha$ where $\delta$ is the Dirac's delta and $p \in ]0;1[$ ...
1
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3answers
91 views

How is the fourier series of $\frac{\pi-x}2$ derived?

$$S = \sum_{n=1}^{\infty} \frac{\sin(n)}{n} $$ I seem to have found on the web: $$\frac{\pi-x}{2}=\sum_{n\geq1}\frac{\sin\left(nx\right)}{n} \space, x \in(0, 2\pi)$$ Then: $$x = \pi - 2\sum_{n=1}^{...
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1answer
46 views

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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1answer
80 views

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 ...
3
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1answer
67 views

Finding $ \lbrace a_{n}\rbrace $ s.t. $\mathop {\lim }\limits_{n \to \infty }a_{n}=1$ and $\mathop {\lim }\limits_{n \to \infty }a_{n}^{n}=2015$

The following problem appears in our analysis assignment. Find a sequence $ \lbrace a_{n}\rbrace $ of real numbers such that $$\mathop {\lim }\limits_{n \to \infty }a_{n}=1\text{ and }\mathop {\...
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0answers
43 views

Re-arrangement of a series

If we define a series as $\sum_{k=1}^{\infty} s_k$, then a series $\sum_{k=1}^{\infty} t_k$ is a re-arrangement of $s_k$. If there is a one-to-one association like a function $f : \mathbb{N} \to \...
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2answers
70 views

Rewriting series.

I need to rewrite the following product of series $$(\sum\limits_{p=0}^{\infty} x^p)( \sum\limits_{q=0}^{\infty} x^{3q}) (\sum\limits_{r=0}^{\infty} x^r)$$ i know how to rewrite the first and the ...
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0answers
45 views

Is it possible to find out how many results were unexpected?

During a school year Andrew was given 40 mathematical problems as part of his assessment, one problem per week. As a result of marking he could receive 2,3,4 or 5 marks for each problem. Andrew called ...
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3answers
2k views

Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequence $(...
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3answers
167 views

A sequence for which the set of limits points is the interval $[0,1]$.

My professor challenge me to give a sequence with limit points from zero to one including 0 and 1?
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3answers
120 views

If $a_n>0$ converges to $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a.$ [closed]

I would appreciate your help! How can we show that if a sequence of positive real numbers $a_n$ converges to $a\in\mathbb{R}$ with $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a$.
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1answer
74 views

How to approximate this summation?

can anyone help me understand how approximate this summation? \begin{align} \frac{r-1}{n} \sum_{i=r}^{n} \frac{1}{i-1}. \end{align} I should be able to get \begin{align} x\int_{x}^{1}\frac{1}{t}dt \...
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1answer
42 views

Proving $\sum_{n=1,3,5..}^{\infty }\frac{\sin^r(n\pi/3)}{n^2}=\frac{3^{0.5r-2}}{2^r}\pi^2$

Proving $$\sum_{n=1,3,5..}^{\infty }\frac{\sin^r(n\pi/3)}{n^2}=\frac{3^{0.5r-2}}{2^r}\pi^2$$ if the $r$ an even integer number greater than 0 I don't have the enough experience to prove formulas ...
0
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1answer
54 views

complex analysis exponential series evaluation

Evaluating the series $$f=\sum \frac{27^n}{(3n+1)!}$$ I could simplify this to $$\frac {1}{3}\sum \frac{3^{3n+1}}{(3n+1)!}$$ and use the chart $$\sum \frac{x^{an+b}}{(an+b)!}$$ to evaluate $f$. But ...
0
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1answer
150 views

Find an equivalent of sequence at infinity

I want to find an equivalent at infinity to those two sequences and then deduce their possible limits: $$ u_n=\frac{(-1)^n+1}{n+\sqrt{n}},\,v_n=\frac{n^5+e^n}{2n+e^n}. $$ For the first one, I found, ...
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1answer
39 views

How do I create a mathematical formula for a finite sequence?

I have this sequence of values and want to create a formula to calculate serially all factors for X values ranging between 1 and 8. Can somebody point me the direction please? ...
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2answers
43 views

Need help comparing this series for a limit test

I'm not sure what to compare the following series to. Would $\dfrac{1}{n^{\frac{3}{2}}}$ work? $$\sum_{n=1}^\infty \frac{n^2-1}{n^4+2n+1}$$
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3answers
136 views

complex series exponential evaluation

How do we evaluate the series $$\sum \frac{27^n}{(3n+1)!}$$ I could simplify this to $$\sum \frac{3^{3n}}{(3n+1)!}$$ but typically the tables provide you with the general series form of $$\sum \frac{...
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3answers
57 views

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 ...
3
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2answers
63 views

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 ...
4
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4answers
85 views

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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2answers
67 views

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? $$ \exists ? (u_n) \;|\; u_n \sim 0 \mbox{ but }u_n\nrightarrow 0. $$ ...
0
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1answer
52 views

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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1answer
99 views

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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2answers
84 views

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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1answer
72 views

Baby Rudin Problem 6(d) in Chapter 3: What about the convergence of this series for $\vert z \vert >1$?

Let $a_n \colon= {1 \over {1+z^n}} $ for $n = 1, 2, 3, \ldots$, where $z$ is a given complex number. Then what about the convergence of the series $\sum a_n$? My effort: When $\vert z \vert \leq ...
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0answers
49 views

$\displaystyle \sum_{1 \le k,j \le n}a_kb_j$ is maximum when $k=j$.

Problem: If $a_r >a_{r+1}>0$ and $b_r >b_{r+1}>0$ for $1 \le r \le n-1$ prove that: $\displaystyle \sum_\limits{1 \le k,j \le n}a_kb_j$ is maximum when $k=j$. I already know a proof of ...
2
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2answers
261 views

Absolute summability, square-summability and $\sum_t(\log t)^2 x_t^2<\infty$, which is the strongest?

I'm working through a few convergence conditions for time series. In particular, I have run into $$ \sum_{t=1}^\infty|x_t|<\infty\tag{$*$}, $$ $$ \sum_{t=1}^\infty x_t^2<\infty,\tag{$**$} $$ $$ \...
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1answer
30 views

Using the limit comparison test to find whether a series is divergent or convergent.

The series is: I'm trying to compare it to: Which converges when using p-series. However I'm not sure if both converge using the limit comparison test.
3
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2answers
312 views

Understanding how to prove limit theorems for sequences.

How do you know what arbitrary value to choose for epsilon as in this document, and when should the triangle inequality be considered when writing your proof?
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1answer
73 views

Uniform convergence of $\sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}}$

Does $\sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}}$ converges uniformly. $-1<x<1$ I have tried to bound it by Weierstrass M-Test but haven't been successful. I have also tried ...
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4answers
61 views

How to know which test to chose when proving the convergence of a serie?

I need to prove the convergence of this serie : $$\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n^2+n}}$$ I tought the easiest would be to use the ratio test, however I can't figure out how to solve. ...
4
votes
3answers
258 views

How to prove the convergence of the series $\sum\frac{(3n+2)^n}{3n^{2n}}$

I need to prove the convergence of this series: $$\sum\limits_{n=1}^\infty\frac{(3n+2)^n}{3n^{2n}}$$ I've tried using Cauchy's criterion and ended up with a limit of $1$, but I already know it does ...
3
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0answers
79 views

Do there exist any cycles for these number sequences?

We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots *p_m^{...
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2answers
440 views

Induction Sequence on Points on a Circle

Suppose that n a’s and n b’s are distributed around the outside of a circle. Use mathematical induction to prove that for all integers n ≥ 1, given any such arrangement, it is possible to find a ...
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3answers
130 views

General term of the series - find

What is the general term of the series: $$-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+...$$ I think that the denominator will be $(n+1)$. But what next?
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1answer
51 views

Estimating distance between two functions

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. I am trying to prove that $F$ is closed in $BC(\Bbb R, \Bbb R)$, where $BC$ is the space of ...
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0answers
47 views

Given sum and sum of squares of numbers in an infinite geometric progression, find $y_1$ and $q$

The sum of the numbers in an infinite geometric progression is $8$. Find $y_1$ and $q$ if the sum of the squares of the series numbers is $\frac{512}7$. This is the way I solved it: $$S=\frac{y_1}{...
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1answer
49 views

proof that this sequence converge if the other converge

If $(a_n)$ and $(b_n)$ are two real sequences such that $\forall n\in\mathbb{N}$ we have $e^{a_n}=a_n+e^{b_n}$ prove that $a_n>0\Rightarrow b_n>0$ prove that if $a_n>0\forall n\in\...
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2answers
27 views

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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0answers
185 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

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^...
3
votes
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?
2
votes
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 ...
5
votes
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 ...
3
votes
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, ...
0
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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,...
2
votes
2answers
82 views

How do I find a closed form for this sequence?

I want to find a closed form for $$\sum_{i=0,1,...}{\left\lfloor\frac{n}{2^i}\right\rfloor}$$ e.g. ...
1
vote
1answer
56 views

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}...