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

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0
votes
2answers
49 views

Decreasing integral sequence

How does one show that $I_n = \int\limits_0^1 x^n e^x dx$ is decreasing? The best I came up with is this: $I_{n-1} - I_n= \int\limits_0^1 e^xx^{n-1}(1-x)dx$, but how do we go from here? I'd ...
0
votes
1answer
27 views

About $\sum_{i\geq 1}\frac{1}{(n+i)_{n+1}}$ and $\sum_{i\geq 1}\frac{1}{i^2-i-1}$

I was playing around with Zeta function and changed it as following to find that $$\sum_{i=1}^{\infty} \frac{1}{i\cdot(i+1)\cdot(i+2)\cdot\ldots\cdot(i+n)} = \frac{1}{n\cdot n!}$$ ...
1
vote
1answer
60 views

What sum to $\sum_{n=0}^\infty\frac{x^n}{(n+1)!}$ in its convergence radus?

My task is this: Find the sum to $$\sum_{n=0}^\infty\frac{x^n}{(n+1)!}.$$ in its convergence radus. My work so far: By ration test we get ...
1
vote
1answer
45 views

Convergence of $\sum_{n=1}^\infty -\frac{1}{n} (\frac{x}{c})^n$

$\sum_{n=1}^\infty -\frac{1}{n} (\frac{x}{c})^n, c>0$ I have proved that this series converges pointwise on $[-c,c)$ by observing that for $x=-c$ the series is the alternating harmonic series, and ...
0
votes
1answer
22 views

Convergence of $\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$

For which $\alpha$ and $\beta$ is the sum $$\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$$ $$µ\left(k\right) = \left\{ \begin{array}{lr} \alpha & : k\ is\ even\\ \beta & : k\ is\ ...
0
votes
0answers
16 views

Is there any closed form of an upper-bound of the following equation?

Could you please let me know if you can find the closed form of the following Equation (or any upper-bound that converges): $\sum_{i=1}^\infty(\dfrac{X}{Y^i})^i i!$, where $0<X<1$ and $Y>1$. ...
0
votes
1answer
26 views

Proving that sums of convergent sequences are complete metric spaces

Let $L_1$ be the set of all sequences of real numbers $$x = (x_1,x_2,..., x_n, ...) $$ with the property that $\sum_{n=1}^\infty |x_n|$ is convergent. If we define $$d_1(x,y) = \sum_{n=1}^\infty ...
1
vote
1answer
101 views

Finding a closed form for $\sum_{k=1}^{\infty}\frac{1}{(2k)^5-2k}$

I'm looking for a closed form for the expression above. I know that Ramanujan gave a closed form for $$ \sum_{k=1}^{\infty}\frac{1}{(2k)^3-2k}= \ln(2)-\frac{1}{2} $$ I wonder if it is possible to ...
2
votes
0answers
23 views

compactness of thes sequence set

Let $S$ be a compact (in the usual topology) subset of $\mathbb R^n$, let $W = \{(q_k)_{k\in\mathbb{N}}\,\mid\, q_k\in S\}$ be the set of all the sequences taking elements in $S$, let ...
1
vote
1answer
38 views

hyperbolic sum and elliptic integral 2

I try to show that $$\sum _{k=1}^{\infty } k^{36} \text{sech}(\pi k)=\frac{41222060339517702122347079671259045}{137438953472}+\frac{i \left(\psi _{e^{\pi }}^{(36)}\left(1-\frac{i}{2}\right)-\psi ...
0
votes
2answers
34 views

How to find limit of the sequence

A sequence is defined by recurrence relation $f_{n+1}=\frac{3}{7}f_n+8$ with $\mu_0=-14$, then what is the limit of the sequence? $14$ $-14$ $\frac{-3}{7}$ $\frac{3}{7}$ My attempt: As wiki ...
0
votes
1answer
43 views

Prove that $\sum_{i=0}^{jk}{jk\choose i}=\left[\sum_{i=0}^{j}{j\choose i} \right]^k$

Prove that, (1) $$\sum_{i=0}^{jk}{jk\choose i}=\left[\sum_{i=0}^{j}{j\choose i} \right]^k=\left[\sum_{i=0}^{k}{k\choose i} \right]^j$$ I can't find something similar hints from mathword (binomial ...
2
votes
1answer
24 views

Series converge/disconverge

I need to prove/disprove this: $1.$ If $\sum a_n$ converge then $\sum a_n*a_{n+1} $ converge. $2.$ if $\sum a_n$ converge then $\sum \frac{a_n*\sqrt{n}}{\sqrt{n}+1}$ converge. Someone can give me a ...
0
votes
1answer
28 views

Find if the series $\displaystyle\sum_{n=1}^\infty\frac{\cos n\sin\frac{1}{n}}{n}$ converges

As far as I understand this is a Leibniz series therefore it's converging. What I was thinking is finding a way to "change" the $\cos n$ to $\cos {\pi n}=(-1)^n$ and get a ...
3
votes
1answer
44 views

Find the interval of convergence to $\sum_{n=2}^\infty\frac{(-1)^nx^{n}}{n(n-1)}.$

My task is to find the interval of convergence to:$$\sum_{n=2}^\infty\frac{(-1)^nx^n}{n(n-1)}.$$ My work so far: Taking $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=|x|<1\implies ...
1
vote
1answer
40 views

If $g(x) = xf(x^2)$ and $f(x)=\sum_{n=0}^\infty \sin(\frac{\pi}{n+2})x^n$, what is $f^{(20)}(0)$ and $g^{(35)}(0)$?

My task is this: (i)Let $$f(x)=\sum_{n=0}^\infty \sin\left(\frac{\pi}{n+2}\right)x^n.$$ Find $f^{(20)}(0)$ and $g^{(35)}(0)$ when $g(x) = xf(x^2)$. (ii)Find ...
0
votes
2answers
24 views

How to determine the convergence radius and intervale of convergence from this sum

I have to find the converge radius and interval of convergence for the serie, I've tried the hHadamard criteria but I had no succes. I hope you can help me. $\sum_{n=1}^{\infty}(2+(-1)^n)(1+x)^{n-1}$ ...
5
votes
1answer
49 views

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: ...
-3
votes
1answer
25 views

Recursive/Explicit Formula, Geometric/Arithmetic Sequences [on hold]

Write an explicit formula for the following arithmetic sequence $-4,-1,2,5, \ldots$ Write an recursive formula for the following arithmetic sequence $-4,-1,2,5, \ldots$ Write an explicit formula for ...
-4
votes
0answers
36 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
2
votes
2answers
103 views

Prove $\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$

We took this idea from Simon Plouffe see here $$\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$$ Can anyone prove this identiy? We found this ...
0
votes
1answer
21 views

$A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
1
vote
0answers
8 views

How to express a homogenous function using an infinitely recursive matrix operation?

As is well known, if the multivariate function $f(\mathbf{x})$ is homogenous of degree $h$, then the partial derivatives of $f$ are homogenous of degree $h-1$. Also, say that we know $f$ is ...
2
votes
2answers
57 views

How to tell if the series $\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4}$ converges?

$$\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4} $$ Trying to figure out if this converges, trying to use the divergence test but I can't figure out how to simplify the problem.
1
vote
1answer
12 views

Order of magnitude and absolute convergence

In connection with studying absolute convergence of Fourier series, I started to wonder about the following: Assume that $$S\sim \sum_0^{\infty}a_n,$$ where all we know about $a_n$ is that ...
1
vote
2answers
31 views

$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$

I need to prove the following: $$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$$ It looks pretty intuitive since I can make $\frac{1}{n}$ as small as I want, thusk making $a$ as close as to ...
0
votes
1answer
23 views

Does this infinite product converge? And how to express it neatly?

First of all, does this product have a "nicer" functional form--i.e., analogous to how you can write geometric sums in a nice closed expression: $$(x-0)(x-1)(x-2)...(x-n)$$ Secondly, does this ...
0
votes
0answers
23 views

Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
5
votes
2answers
85 views

Is the series convergent

Is series $\sum_1^\infty \frac{\ln(1+1/2) \ln(1+1/4) \cdots \ln(1+1/(2n))}{\ln(1+1/1) \ln(1+1/3) \cdots \ln(1+1/(2n-1))} = \sum_{n=1}^\infty \prod_{m=1}^n \ln(1+1/(2m))/(\ln(1+1/(2m-1))$ convergent ?
1
vote
0answers
36 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
0
votes
3answers
24 views

Rearrangement of alternating harmonic series that does not converge

From Riemann's series theorem, we know that, given a conditionally convergent series, we can permute the elements of the series in order to basically do whatever we want. I have seen a rearrangement ...
0
votes
2answers
34 views

Proof series decreases by induction

I have a question regarding a proof by induction. We have to see whether or not the following series converges. $$U_n = \frac{1 \cdot 4 \cdot 7 \cdots (3n - 2)}{2 \cdot 5 \cdot 8 \cdots (3n-1)}$$ I ...
0
votes
0answers
29 views

Proving absolute or conditionally convergence

I have the following partial series - $$ \sum_{n = 0}^\infty {a_n} $$ when $$a_n=\begin{cases} \frac{1}{n} &; \quad n \ \text{is even}\\ \frac{-1}{n^2} &; \quad n \ \text{is odd}\ . ...
1
vote
3answers
70 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
-1
votes
1answer
19 views

Limit of a sequence an and bn [on hold]

I have two sequences $a_n$ and $b_n$ and the limit of $a_n$ is $a \in \mathbb{R>0}$ and the limit of $b_n \to \infty $. Now I have to show that $a_nb_n \to \infty$. I can imagine that it this is ...
0
votes
1answer
25 views

Problem with the inverse expansion

Let $q=e^{2\pi i z}$ and $t=q-12q^2+66q^3-220q^4+495q^5-...$ Then why is the inverse expansion equal to $q=t+12t^2+222t^3+...$? I also do not understand the notation here: $t$ means $t(z)$ or $t(q)$? ...
-3
votes
2answers
24 views

Determine whether the series is covergent or divergent [closed]

Test the covergence ofthe series $\sum_{0}^{\infty}\frac{5^n+5}{3^n+2} $
1
vote
2answers
38 views

Series converge/converge absolutely/diverge

I need to determine if the two series are converge/converge absolutely/diverge: $1.\sum^{\infty}_{n=2}\frac{\sin{n}}{n\sqrt{n}}+\frac{\cos{n}}{n\ln(n)} $. ...
4
votes
1answer
40 views

Show that if $(\sum x_n)$ converges absolutely and $(y_n)$ is bounded then $(\sum x_n y_n)$ converges

This is the exercise 2.7.6 of the book Understanding analysis of Abbott, I want a check of my proof and if is needed additional information to complete it. a) Show that if the sequence $(\sum ...
1
vote
1answer
34 views

Sequence existing for a set of conditions

Let function $f$ is continous and limited on the interval $(x_0, +\infty)$. Prove $\forall \ number \ T \ \exists \ sequence \ \{x_n\},\ \lim_{n \to\infty}{\{x_n\}} = +\infty $: ...
0
votes
1answer
41 views

$\frac{\sin(nx)}{n^p}$ series convergence check

Check for which $p$ functional series $\sum {\sin(nx) \over n^{p}}$ converges. It is easy to see that for any p > 1 it does converge for any real x. Because of comparsion test. How to figure out other ...
0
votes
1answer
31 views

Subsequence and diagonal process

We consider a sequence of functions défined on $\mathbb R^n$ by $f_m(x)=f(\frac{x}{m}),\ \forall m\in \mathbb{N}$ such that : 1) $f=1 $ in $B(0,1)$ 2) $\mathrm{supp\,} f\subset B(0,2)$ 3) $f ...
0
votes
2answers
43 views

how many n-digit palindromes exist

I'd never seen this kind problem before, and don't know where to start. Any help is appreciated. Thank you very much! A palindrome is a number that is the same forwards and backwards. For example, ...
0
votes
0answers
44 views

Series $\sum \lambda^{n-k} c_k $ converges to zero

Let $(c_n)$ a sequence of real number, such that $\lim_{n \to \infty} c_n=0$, Let $0<\lambda<1 $ and $\lambda^nc_0+\lambda^{n-1}c_1+\cdots+\lambda c_{n-1}+c_n=y_n$ a sequence. I have to prove ...
1
vote
2answers
33 views

Limit of a sequence, possibly requires epsilon delta

Show that if $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ are sequences for which $\lim_{n\to\infty} a_n = 0$ and $\{b_n\}$ is bounded, then $\lim_{n\to\infty} a_nb_n=0.$ This is what I ...
13
votes
0answers
74 views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
1
vote
1answer
57 views

How to find the sum of the series $\sum_{n=1}^\infty \frac{2^n+8^n}{11^n}$?

$$\sum_{n=1}^\infty \frac{2^n+8^n}{11^n}$$ My textbook does not give any example to help solve problems like this, only geometric series and I do not believe this is a geometric series?
3
votes
1answer
34 views

Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$

I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$. The ...
6
votes
2answers
300 views

Is it bad to call series a generalization of sum?

In a recent question I asked why series has a name separate from that of sum, and the general answer was that a series does not have the nice properties of sum. Does this mean it is bad to call series ...
0
votes
0answers
68 views
+100

Are there bounded nonconvergent sequences satisfying this recurrence relation?

In this question, we ask about convergent sequences $(p_n)$ satisfying $p_{i} = 2p_{i+6} - {1 \over 2}p_{i+1} - {1 \over 4}p_{i+3} - {1 \over 8}p_{i+7} - {1 \over 16}p_{i+15} - {1 \over 32}p_{i+31} - ...