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

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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
vote
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
votes
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
votes
3answers
126 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
0
votes
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
votes
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
votes
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
vote
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
vote
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 ...
-2
votes
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
vote
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
votes
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
vote
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
votes
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 \{ ...
0
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
vote
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
votes
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 ...
1
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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
votes
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
votes
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) = ...
-1
votes
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
votes
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?
0
votes
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
votes
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
votes
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
vote
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
votes
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 ...
4
votes
4answers
110 views

Finding out a limit using Taylor series.

So the limit is the following: $$\lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}}$$ Expansions for $\sin(x)$ and $\cos(x)$ are given: $$\sin x = x-\frac{x^3}{3!} + ...
1
vote
1answer
36 views

Show that $f_n(x) = \frac {2x+n}{x+n}$ converges uniformly to $f(x) =1$.

Question: Let b ∈ $\mathbb R$ and let $f_n(x): [-b,b]\rightarrow\mathbb R $ be defined by $f_n(x) = \frac {2x+n}{x+n}$ for all n ∈ $\mathbb{N}$. Show that ($f_n$) converges uniformly to $f$ on ...
0
votes
1answer
33 views

What is the example called, where someone was wrongly convinced of a sequence function because of naive induction.

I remember I have seen a classical example of a mistake, where someone was convinced that a sequence defined somehow had a close form, which did in turn work until some very high $n$. I think the ...
3
votes
1answer
66 views

Find limit recursion of sequence $x_{n+1} = \frac{x_n+ n x_{n-1}}{n+1} $

Prove sequence $$x_{n+1} = \frac{x_n+ n x_{n-1}}{n+1} $$ $$x_0 = 0, x_1 = 1 $$ converges and find it's limit My attempt Let's prove $0 \le x_n \le 1$: $x_n \ge 0 $ (obvious) By ...
1
vote
4answers
68 views

Convergence of logarithmic sum

Numerical experimentations strongly suggest that the series $$\sum_{n=2}^{\infty}\frac{1}{n}\log\left(\frac{n}{n-1}\right)$$ converges and the limit is $\approx0\mathrm.7885$. Could someone give me a ...
-5
votes
0answers
15 views

definition of a limit in terms of sequences and epsilon delta [closed]

link to the question i) for every sequence Xn in the real number system with limit negative infinity, we have limit f(Xn)=infinity n->infinity ii) for any M>0 and N<0 there exists a a> N such ...
-7
votes
0answers
24 views

series convergence help using tests [closed]

Use any theorems or properties of series. This was a question on my homework and I received 0 points. I need help with the entire question. I originally tried to compare part (a) to the ...
1
vote
0answers
13 views

unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...
-1
votes
3answers
61 views

Find the least value of $n$ for which the sum $1+4^2+4^4+\cdots + 4^n$ is greater than $1000$ [closed]

I could not understand the below problem.Can someone help me out. the least value of $n$ for which $1+4^2+4^4+\cdots +4^n$ terms is greater than $1000$?
1
vote
1answer
20 views

Convergence of a sequence to its eigenvector

Consider that we have a matrix $~~~ M \in \mathcal{M}_{n\times n}(\mathbb{R}) ~~~$ symmetric, positive definite. If I set the recurrence relation $$x_{n+1} = \frac{Mx_{n}}{\lVert Mx_{n}\rVert} ...