For questions about recurrence relations, convergence tests, and identifying sequences

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2
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0answers
17 views

Convergence of a series of integrals

Definition of problem I would like to find the large $N$ behaviour of a summation series and specifically the conditions under which it converges. I have found one condition, but I think it might be ...
1
vote
1answer
22 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
0
votes
3answers
20 views

Show convergence of 1/cosh series

Sorry if my english is not correct. Feel free to edit and ask questions. I need to test the following series on convergence: a) $$ \sum_{n=0}^{\infty}\frac { sinh(n) }{ e^n } $$ and b) $$ ...
1
vote
1answer
30 views

Exact value of a sum involving harmonic numbers

Could somebody tell me the exact value of this series? $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n} $$ Thanks!
0
votes
0answers
10 views

How to use finite differences to find approximate functions for a set of data?

Given the data: x: 0 2 4 6 8 10 12 14 16 y: 5 -8 -11 -9 4 23 52 89 131 How would you use finite differences to find the approximate function to model this data? There is no common ...
7
votes
1answer
73 views

Limits, Taylor expansion

Find the limit: $$ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} $$ for $n$, $m$ natural ...
1
vote
0answers
11 views

Complementary Golay sequences and sum of their autocorrelation function

Golay complementary sequences are aperiodic sequences made up of +1 and -1 that have nice property which is that their autocorrelation that sum up as korneckr delta function. Example $G_{a4}=(+1, +1, ...
1
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0answers
33 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
1
vote
1answer
28 views

Power series representation?

The function to represent as a power series is: $$ \frac {10} {(x-10)^2} $$ Any help is, as always, appreciated.
0
votes
1answer
24 views

Interval of convergence of power series?

If the power series is: $$ \sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n+1}} $$ and I've found the interval to be $$ -1 < x < 1 $$ then would the answer $$ (-1, 1) $$ work? some other options ...
1
vote
0answers
33 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
2
votes
3answers
29 views

(Simple question) Radius of covergence of power series?

For the power series: $$ \sum\limits_{n=1}^{\infty}\frac{(x-1)^n}{2^n} $$ Would radius of convergence be $$ x = 1 $$ ?
2
votes
1answer
41 views

What is the center of power series?

The power series is: $$ \sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1} $$ Any help appreciated!
0
votes
0answers
19 views

$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
1
vote
1answer
12 views

Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
1
vote
0answers
17 views

Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
2
votes
2answers
55 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
0
votes
1answer
44 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
2
votes
2answers
31 views

Continuity and differentiability of $f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!(x+n)}$

Given the series: $$\sum_{n \ge 0} \frac{(-1)^n}{n!(x+n)}$$ Let $f_n(x)$ denote its general term. Let $f(x)$ denote its sum (when exists). The question asks to: $i)$ Find the domain $\mathbb D$ on ...
0
votes
1answer
42 views

Proving a series from zero to infinity is half of a series from minus infinity to infinity?

I want to prove that $$\sum_{n=-\infty}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ is equivalent to $$\sum_{n=0}^{\infty} \frac{2(-1)^n}{(2n+1)^3}$$ I have played around with it and I know that it is correct ...
-4
votes
1answer
21 views

Absolutely convergent, conditionally convergent or divergent [on hold]

I have this question: $$\sum_{n=1}^\infty \frac{\cos\left(\frac{n\pi}{12}\right)}{n\sqrt n} $$ How do I figure out if it's absolutely convergent, conditionally convergent or divergent?
3
votes
2answers
75 views

If $\frac{a_{n+1}}{a_{n}} \nearrow 1$ when $n \to \infty$, does $\sum_{n=1}^{\infty} a_{n}$ converge?

Suppose $(a_{n})$ is a sequence which satisfies $a_{n} > 0, \forall n \in \mathbb{N}$. The ratio test states that if $\frac{a_{n+1}}{a_{n}} \to L < 1$ when $n \to \infty$, then the series ...
0
votes
0answers
52 views

Finding the limit of this specific series

So, I have to calculate: $$\lim _{ n\to\infty } \prod_{k=2}^{n} \Big(2-\sqrt[k]{2}\Big)$$ So far I managed to get to: $$\lim _{ n\to\infty } \sum_{k=2}^{n}\Big(1-\sqrt[k]{2}\Big)$$ Any help will ...
0
votes
1answer
34 views

Example of a convergent series for which integral test fails?

Is there example of a convergent series for which integral test fails or can not be applied? Just wondering if integral test is the silver bullet of convergence tests, or are there any series that any ...
3
votes
0answers
22 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k^n}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
0
votes
3answers
31 views

Determine the greatest value of $n$ for which $b > a$

Let $a_n$ and $b_n$ be two recursive sequences so that: $a_{n+1} = a_{n} + 2000$ where $a_{1} = 500$ and $b_{n+1} = \frac{b_{n}}{1,001}$ where $b_{1} = 50000$ Determine the greatest value of $n$ for ...
2
votes
1answer
25 views

Bounding infinite series derived from polygamma functions

Let $f(x) = 2 \psi^{(1)}(x+1) + x \psi^{(2)}(x+1) $ for $ x > 0 $, where $\psi^{(i)}(x)$ is the $i^{th}$ derivative of the digamma function $\psi(x)$. The goal is to prove that $ f(x) < ...
-1
votes
1answer
27 views

Find the point-wise limit of this sequence of function $\{f_n(x)\}$.

Consider the sequence of function $\{f_n(x)\}$ in $[0,1]$ where , $$f_n(x)=\begin{cases}0 & \text{ if } x=0\\n^2x & \text{ if } x\in [0,\frac{1}{n}]\\-n^2x+n^2 & \text{ if } x\in ...
1
vote
0answers
38 views

There exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$

Suppose $f$ has an Essential Singularity at $Z_0$. Then there exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(Z_n)| = \infty$ Here two cases arise If there exist a nbd ...
0
votes
2answers
25 views

Sums of converging limits

How can I prove the property that if the sequences, $(x)\rightarrow x' $ and $(y)\rightarrow y'$ then $(x) + (y)\rightarrow x'+y'$
-6
votes
2answers
58 views

Is $(-1)^{n!}$ convergent? [on hold]

I don't think I can use the alternating series test because of the factorial sign, but I don't know how else to solve this. can you please give any hints ?
0
votes
1answer
47 views

How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$

For reals $x \ge 3$, let $f(x)$ denote the function $f(x) = \frac{-x + x\sqrt{4x - 3}}{2}$. Now suppose that $a_1, a_2, \ldots, a_{2013}$ is a sequence of real numbers such that $a_1 > 3, a_{2013} ...
1
vote
0answers
29 views

Show that a sequence is between a range

I got this question in class which I'm having trouble proving I tried investigate the sequence a little bit but it doesn't seem like I'm doing the right think, some help? $ \frac{39}{e^2} \le ...
0
votes
1answer
27 views

Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$

I have this question which I'm having trouble solving, can I use some help? :) Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$: $ \sum_{n=1}^{\infty} ...
1
vote
3answers
54 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
0
votes
2answers
34 views

Error in a Maclaurin series

I'm having trouble figuring out what I have to do with this question. "Using Taylor's theorem, determine the largest positive real value $r$ for which we can guarantee that the Maclaurin polynomial ...
4
votes
2answers
52 views

Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$

If $a_{n}$ is a sequence and $|a_{n+1} - L| < |a_{n} - L|, \forall n \in \mathbb{N} $, then clearly the sequence $s_{n} = |a_{n} - L|$ converges (it's decreasing and bounded by $0$). Does it ...
0
votes
1answer
43 views

Closed form for series involving harmonic numbers

Is there a closed form for this series values: $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(n)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^k \frac{1}{i^n} $$ and n is a positive integer. Thanks!
2
votes
3answers
144 views

Determining the limit of this series

$$ \sum_{n=0}^\infty \frac{(-2)^n + 2^{3n}}{3^n4^n} = $$ $$ \sum_{n=0}^\infty \frac{(-1)^n2^n}{3^n4^n} + \sum_{n=0}^\infty \left(\frac{2}{3}\right)^n = $$ $$ \sum_{n=0}^\infty (-1)^n\frac{1}{6^n} + ...
1
vote
0answers
42 views

Is the Ulam Spiral just a coincidence?

I was messing around with the Ulam spiral because I was a little skeptical on it having any actual relevance. I noticed that if you lay out the spiral and then circle all the even numbers, it displays ...
1
vote
0answers
32 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
-2
votes
1answer
25 views

Subsequence $x_{nk}$ tend to ∞? [on hold]

We have a sequence $x_n$ where $n\geq1$ so that $x_n$->∞.
-2
votes
1answer
42 views

Existence of divergent series $\sum_{n=1}^ \infty a_n$ of real numbers whose partial sums are bounded and $\lim (na_n)=0$ [on hold]

Does there exist a sequence $(a_n)$ of real numbers such that $\lim_{n \to \infty} (na_n)=0$ , the partial sums of $\sum_{n=1}^ \infty a_n$ are bounded , but $\sum_{n=1}^ \infty a_n$ is divergent ? ...
0
votes
1answer
25 views

Could the Hamel basis of $\mathbb{R^Z}$ be the set $\mathbb{R^Z}-{\mathbf{\{0\}}}$?

This is the follow up question to this question (*) According to page 2 of this link 1 and this link 2, $\mathbb{R^Z}$ (which is referred as $\mathbb{R^\infty}$ in link 1) has elements of the ...
2
votes
1answer
52 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
vote
1answer
78 views

Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
-6
votes
0answers
19 views

Ples help with this series [on hold]

1/(n^(3) +2n)^(1/2) it converge or diverge
2
votes
1answer
98 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
2
votes
6answers
102 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
-2
votes
1answer
38 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.