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

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1answer
74 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 ...
1
vote
0answers
42 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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votes
2answers
67 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 ...
4
votes
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
143 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
119 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
73 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 ...
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1answer
51 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
141 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
135 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 ...
4
votes
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
58 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
votes
4answers
84 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
66 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? $$ \exists ? (u_n) \;|\; u_n \sim 0 \mbox{ but }u_n\nrightarrow 0. $$ ...
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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 ...
1
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1answer
98 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 ...
2
votes
2answers
78 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
67 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
votes
2answers
215 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
29 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
285 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?
1
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1answer
65 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 ...
1
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4answers
59 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
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3answers
255 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
76 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 ...
1
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2answers
426 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 ...
2
votes
3answers
112 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?
1
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1answer
50 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
46 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: ...
0
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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 ...
0
votes
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 ...
4
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0answers
176 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: ...
3
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4answers
108 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
88 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
79 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
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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, ...
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1answer
84 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, ...
2
votes
2answers
80 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
53 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 ...
0
votes
5answers
52 views

Find general formula for a series

Having the following series: $$ - \frac{1}{2}+\frac{1}{6}- \frac{1}{10}+\frac{1}{14}-\frac{1}{18}+\ldots$$ What is the easiest approach to find a general formula for this series?
1
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4answers
162 views

Investigate the convergence of

Investigate convergence of the following series: $$\sum_{n=0}^\infty\left( \frac{2+(-1)^n}{\pi} \right)^n$$ Which convergence criterion shoul be applied?
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votes
3answers
65 views

Limit of $\frac{2 \sqrt[n]{n!}}{n}$ [duplicate]

Why the limit of the following sequence is like this: $$\frac{2 \sqrt[n]{n!}}{n}=\frac{2}{e}$$
0
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1answer
73 views

Series expansion of $\exp(-x)$ without alternating terms?

Is there a series expansion of $\exp(-x)$ that does not have alternating terms? Sorry for the ambiguity. I want a power series, but since powers of $x$ blow up as $x\to \infty$ and powers of ...
1
vote
1answer
87 views

Proving uniform continuity of sequence of functions

Lets take the infinite sequence of functions $f_{n}(x) = x/(x +1/n) , x \in [0, 1], n \in \Bbb N$. Show that each function $f_{n}$ is uniformly continuous. My solution: Given $\epsilon >0$, let ...