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

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0answers
21 views

show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
2
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1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
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0answers
40 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
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1answer
21 views

Sum of product of convergent series

Suppose $\{a_n\},\ \{c_n\}\subset\mathbb{R}$ satisfies $\lim\limits_{n\to\infty} a_n = a\in \mathbb{R}$, $\lim\limits_{n\to\infty} c_n = 1$. Prove that $$ \lim\limits_{M\to\infty} ...
0
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1answer
96 views

Is there any summation method that assigns $ \sum_{n=1}^\infty \frac{1}{n} =-\frac{\pi}{2}$

I don't know too much about alternate summation methods, but am interesting to know if any give the sum of the harmonic series to be $$-\frac{\pi}{2}$$
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2answers
53 views

$\sum_{k=1}^n 1/k - \log n$

I got this question : $$a_n = \sum_{k=1}^n \frac 1k - \log n$$ I proved that $\lim a_n $ exist. Now I have to prove: $$ 0<a_n-\lim a_n\le \frac 1n $$ for every $n \in \mathbb N$. I tried ...
0
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1answer
43 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
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1answer
51 views

If $A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}$, find $\lim_{n \to \infty} A_n$

Set $$A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}.$$ Find $\displaystyle \lim_{n \to \infty} A_n$. I tried solving this using a reduction method. That is, reducing the above series to an ...
0
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1answer
26 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...
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1answer
39 views

Alternating Euler sums with even index

We are all aware of the generating function of $\frac{x \arctan x}{x^2+1}$ which is: $$\frac{x \arctan x}{x^2+1} = \sum_{m=1}^{\infty} (-1)^m \left ( \mathcal{H}_{2m} - \frac{1}{2} \mathcal{H}_m ...
0
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1answer
38 views

Investigate convergence of $\sum_{n=2}^\infty \frac{1}{n(n-1)}$

Investigate convergence of: $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ Now I know by $p$-series test this summation converges however, is there a way to prove that this series converges by some ...
1
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2answers
25 views

Solving inequality with a recursive formula without its closed form?

I have the following problem: $$a_{1}=1$$ $$a_{2}=3$$ $$a_{n+2}=a_{n+1}+5a_{n}$$ I have to prove this inequality: $$a_{n}<1+3^{n-1}$$ So my question, is there a way to solve this inequality without ...
3
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2answers
49 views

Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
1
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1answer
30 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
3
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3answers
126 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
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1answer
22 views

Show the following series converges uniformly using Weierstrass M Test

I'm trying to show that the following series converge uniformly by using the Weierstrass $M$ Test. $$ \sum ^{\infty}_{j=0}z^{n},\ \ \ 0\leq \left | z \right |< R,\ \ \ R<1 $$ and $$ \sum ...
3
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2answers
35 views

$\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$

I must prove: $\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$ Well, I know that $$\lim x_n = a \implies |x_n-a|<\epsilon$$ $$\lim \frac{x_n}{y_n} = b \implies ...
0
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1answer
34 views

Is there a general approach to find an explicit formula to a recursive sequence?

I've been doing a bunch of exercises where I need to find the explicit formula for a given sequence... $$a_{n+2} = 6 a_{n+1} - 9 a_{n}$$ The first were easy with $a_{0} = 1$ and $a_{1} = 3$ i.e.. But ...
1
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1answer
22 views

$a_n, b_n$ bounded, $a_n+b_n=1$,$z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$

I must show that if $a_n, b_n$ are bounded such that $a_n+b_n=1$, and if $z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$ My idea was: $$(a_n+b_n)(z_n+t_n) = a_nz_n+a_nt_n+b_nz_n+b_nt_n$$ I ...
1
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1answer
15 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
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0answers
22 views

Finding the nth term of a geometric series [closed]

Please help me find the nth term of the series: $$\frac{2}{5}-\frac{6}{5^2}+\frac{10}{5^3}-\frac{14}{5^4}+\cdots$$
1
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1answer
24 views

Write the series using sigma notation: $f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +\cdots$

$$f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + \cdots$$ I don't know how to get the signs to work like negative, then positive. I have tried to make it like the following: $(-1)^{n-1} \frac{x^{2n-1}}{ ...
0
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1answer
33 views

How to prove $\sum n/3^n$ converges without ratio test?

The only tests my class has learned so far and is allowed to use are: Divergence, Integral, Comparison, Geometric/Harmonic/Telescopic. I have proved that the series converges, via a ratio test, but my ...
1
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3answers
56 views

$\lim a_n = L \implies \lim a_n^2 = L^2$

I have to prove the following: $$\lim a_n = L \implies \lim a_n^2 = L^2$$ I know that $\lim a_n = L \implies \forall\epsilon>0 \ \exists n_0$ such that $n>n_o\implies |a_n-L|<\epsilon$ I ...
0
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0answers
18 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
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1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
3
votes
2answers
44 views

Sum of the series $\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+…$

If $|x|<1$, find the sum of infinite terms of following series: $$\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+....$$ Could someone give me hint to solve this ...
0
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1answer
26 views

Show ${f_n(x)}_ {n=1, \cdots, \infty}$ converges to $0$ uniformly on $(0,1)$

If $f_n(x) = \dfrac{x}{1+nx}$, show that ${f_n(x)}_{n=1, \cdots, \infty}$ converges uniformly to $0$ on $(0,1)$. Here is what I have so far: Let $\epsilon>0$ be given. Pick any $n \in ...
2
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0answers
29 views

Evaluation of an infinite double sum

First of all, my apologies for this trivial question. But I keep getting the wrong answer. I hope someone can point out my error. Let $\lvert \theta \rvert > 1$. The double sum that I have is as ...
0
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1answer
37 views

Proving limit of the sequence ${ x }_{ n+1 }=\frac { 1 }{ 3 } \left( 2{ x }_{ n }+\frac { a }{ { x }_{ n }^{ 2 } } \right) $

Here is the question I should prove. Given: $${ x }_{ n+1 }=\frac { 1 }{ 3 } \left( 2{ x }_{ n }+\frac { a }{ { x }_{ n } ^{ 2 } } \right),{ x }_{ 0 }>0, n \epsilon \mathbb{N} $$ prove ...
0
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1answer
28 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
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5answers
52 views

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$? It was told to me that the series does converge for all $x$, however I have investigated with a computer ...
1
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0answers
41 views

A variant of the exponential integral

Consider the following integral (for $x,y\in \mathbb{R}_{>0}$) $$E(x,y) = \int_0^1 \frac{\mathrm{e}^{-x/s-ys}}{s}\,\mathrm{d}s,$$ which is a variant of the usual exponential integral $E_1(x)$ to ...
0
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1answer
24 views

I am confused on nth term rules / sequences

So ...... I am really bad at maths and I need a bit of help please can you help me complete this and tell me answers The nth rearm of a sequence is n(n-1) 1.what are the first four terms I don't ...
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3answers
36 views

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$. I do not know if that's so easy that I'm simply missing something, but I can't find any criterion which ...
2
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1answer
37 views

$0\leq u_n\leq \frac {1}{n^2}\sum_{k=1}^nu_k\implies $ $\sum u_k$ converges.

Let $(u_n)_{n\geq 1}$ be a sequence of real numbers such that $\displaystyle \forall n\geq 1, \;\;\;\;0\leq u_n\leq \frac {1}{n^2}\sum_{k=1}^nu_k$. Prove that $\sum u_k$ converges. Let ...
0
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2answers
41 views

How to solve this series :$\sum_{k=0}^{\frac{n}{2}}n-k$

I tried to solve this series as follows ; $\sum_{k=0}^{\frac{n}{2}}n-k$ : $ =(\frac{n}{2}+n)+(\frac{n}{2}+1+n-1)+(\frac{n}{2}+2+n-2)+...+(\frac{n}{2}+k+n-k) = ...
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1answer
24 views

Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
-2
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1answer
42 views

How do we show that the sequences is unbounded? [closed]

how do we prove that if $lim_{n \rightarrow \infty}$ $s_{n}/n = L$, $L \neq 0$ then the sequence $s_n$ is not bounded?
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1answer
44 views

Solve $T(n)=16T(n/2)+2n^4$

Solve using the iteration method: $$T(n)=16T(n/2)+2n^4$$ My attempt: $$\begin{align} T(n) &=16\cdot16T(n/2^2)+2(n/2)^4+2n^4 \\ &=16\cdot16\cdot 16T(n/2^3)+2(n/2^2)+2(n/2)^4+2n^4\\ ...
6
votes
1answer
54 views

Show $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in n for $p \in (0,1)$

I need to show that $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in $n$ for $p \in (0,1)$. My attempts have involved trying to show $\begin{eqnarray*} \frac{1-p^{n+2}}{1-p^{n+1}} ...
0
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0answers
18 views

Existence of convergent sequence in a closed set?

Is it true that if $C$ is a closed set, and $x \in {C}$ then there exist a sequence $\{x_i\}$ which tends to $x$? I'm not sure about the correctness of this statement, and whether it is true for ...
0
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0answers
26 views

Does Cauchy's Multiplication Theorem Apply to Formal Power Series?

Suppose we have two formal power series: $$X(z) = \sum_{n=1}^{\infty} x_n{z_1}^n$$ $$Y(z) = \sum_{n=1}^{\infty} y_n{z_2}^n$$ Then $$\mathrm{exp}(X(z)) = \sum_{n=0}^{\infty} \frac{B_n(x_1,x_2, ...
3
votes
1answer
77 views

Proving $\sum_{s \in S} \frac{1}{n}$ converges for $S = \{ s \in \mathbb{N} : s$ has no zeros on its decimal representation $\}$

Consider $S \subset \mathbb{N}$ as the set of numbers which do not have the algarism $0$ on its decimal representation. For instance: $$S=\{1,2, \dots, 9, 11, 12,\dots, 19, 21, 22, \dots\}$$ I want ...
0
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0answers
16 views

Example of oscillatory sequence with sum alternating between $\pm \infty$

I am looking for an example of a sequence $\{a_n\}$ whose sum oscillates between $\pm \infty$. Is it possible to use Alternating series theorem to deduce that this behaviour. The theorem says that if ...
0
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2answers
22 views

What is the shortest progression?

An arithmetic progression (e.g. 1,4,7,10...) can be infinitely long. But how short can it be? Is the sequence 1, 4 still considered a progression?
1
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1answer
23 views

Does the pth James space Jp contain a norm-1 basic sequence domimated by l2? Equivalently, is there a noncompact operator from l2 into Jp?

The $p$th James space, denoted $J_p$, is just the regular James space using the $p$-norm in place of the 2-norm. See here for a complete definition. To use their notation, let $\mathbb{N}_0$ denote ...
1
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0answers
43 views

Does this function have a dense graph?

Let $\mathbb Q =\{q_n:n\in\mathbb N\}$ be an enumeration of the rationals. Let $f(x)=\mid\sin(1/x)\mid$ if $x\neq 0$ and $f(0)=0$. Let $g(x)=\sum_{n=1} ^\infty \frac{f(x-q_n)}{2^n}$. Question: ...
0
votes
1answer
37 views

Shouldn't all alternating series diverge by the diverge test?

An Alternating Series, as defined in my textbook, is of the form $\sum (-1)^n b_n$. If we look at the nth term, the series doesn't appear to converge. If n is odd, the nth term is negative; if it's ...
3
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0answers
38 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 ...