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

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0
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1answer
51 views

How simplify this particular sum?

Can we simplify the following sum? $$\sum_{i=1}^n \binom{n}{i} {(-1)^{i+1}\over 1-2^{-i}}$$ Thank you.
0
votes
1answer
46 views

How to show uniform convergence of series

Let $$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k$$ for $t \in (0,\infty)$, where the $u_k$ is such that $\sum \sqrt{k}u_k$ converges, but we know nothing about the convergence of $\sum ku_k$. How do ...
0
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0answers
12 views

Using Taylor series of function f1(x) in Taylor series of function f2(x) defined in open discs D1 and D2 when D2 lies inside D1

We have two open discs, D1 and D2, whose centres are C1 and C2 respectively. The Taylor series of function f1(x) is defined in open disc D1 while the Taylor series of function f2(x) is defined in open ...
9
votes
2answers
183 views

An infinite series in polygamma function

I'm interested in $2$ things: $1)$ if you're used to such series and when you met before such series and $2)$ the tools you might like to employ $3)$ I don't ask for a solution. Calculating in closed ...
1
vote
1answer
14 views

Showing the following sequence is monotone decreasing

Let $T$ be fixed and define the functions $$a_k(t) = \frac{e^{\mu_k (T-t)} - e^{-\mu_k(T-t)}}{e^{\mu_k T}- e^{-\mu_k T}}$$ for $t \in [0,T]$. Given that $\mu_k$ is a monotonically increasing ...
0
votes
2answers
54 views

Find radius of convergence for the given sequence: $\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$

I've been trying to realize how to find the radius of convergence for this sequence: $$\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$$ I know that it converges for any given $x$, but can someone explain me ...
5
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2answers
58 views

Show that the sequence $\langle b_n\rangle$ Converges to $1$

The following question was asked in my Masters entrance examination but unfortunately I was unable to answer this. Please tell me how to approach this problem correctly. Suppose $\langle ...
0
votes
1answer
38 views

Geometric series in closed form.

$$f=\sum_{k=3}^{\infty}(-1)^kx^{(2k-2)}$$ i would like to write this as the geometric power series! Is there a ritual you have to do to solve this? thanks in advance.
4
votes
3answers
85 views

How to find sum of the infinite series $\sum \frac{1}{ n(2n+1)}$

$$\frac{1}{1 \times3} + \frac{1}{2\times5}+\frac{1}{3\times7} + \frac{1}{4\times9}+\cdots $$ How to find sum of this series? I tried this : its nth term will be = $\frac{1}{n}-\frac{2}{2n+1}$ after ...
0
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0answers
20 views

Convergence of a geometric series

so I'm working on Laurent Series, and got into trouble calculating such series. The problem is taken from here (P. 2, example 2.3): http://sym.lboro.ac.uk/resources/Handout-Laurent.pdf Find ...
4
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2answers
73 views

$a_n\downarrow 0, \sum\limits_{n=1}^{\infty}a_n=+\infty, b_n=min\{a_n,1/n\}$, prove $\sum b_n $ diverges.

$a_n\downarrow 0, \sum\limits_{n=1}^{\infty}a_n=+\infty, b_n=min\{a_n,1/n\}$, prove $\sum b_n $ diverges. In fact, I have known that two positive divergent series $\sum a_n ~\sum b_n$, ...
8
votes
1answer
104 views

Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$

Deteremine the sum of the series $$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$$ So I first write down the $n^{th}$ term $a_n=\frac{\frac{n(n+1)}{2}}{n!}=\frac{n+1}{2(n-1)!}$. So from there ...
1
vote
2answers
47 views

If $\sum a_{2k}$ exists then $\sum a_m$ exists?

Let $\{a_0,a_1,a_2...\}$ be a sequence of real numbers let $s_n=\sum a_{2k}$. If $\lim_{n\rightarrow \infty} s_n $ exists then $\sum a_m$ exists. Is it true? I don§t find a counter example
5
votes
3answers
61 views

Determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$

I tried to use D'Alambert theorem to determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$ . $\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} ...
0
votes
1answer
20 views

Are these statements equivalent?

Statement 1: A sequence $z_n$ is null if $|z_n|<\epsilon$ where $\epsilon$ is postive and arbitrarily small. $n>\mu$ where $\mu$ is positive and arbitrarily large. Statement 2: A ...
0
votes
1answer
53 views

Determine convergence of the series $\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$

How to determine convergence of the series: $$\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$$ I spent most of the time using the Integral criteria (since the function $f(x)=\frac{1}{\ln(x)^{\ln(x)}}$ ...
0
votes
1answer
42 views

Convergence of $\sum \sin(an)z^n$ when $|z| = 1$

Suppose $z \in \mathbb{C}$ and $a>0$ is not an integer multiple of $\pi$ (to avoid the trivial case). $\sum \sin(an)z^n$ has radius of convergence $r=1$ , because $$|\sin(an)z^n| \leq |z^n|$$ So ...
0
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1answer
36 views

Sequence of numbers…is there a shorter way to do this?

This is the exercise: The sequence of numbers $a_1, a_2, a_3 \dots a_n \dots$ is defined as $a_n = \frac 1{n+1} - \frac 1{n+2}$, for each integer $n\ge 1$. What is the sum of the first 15 terms of ...
3
votes
0answers
41 views

Why does $\sum_{n = 0}^\infty \left[ \frac{1}{(2z + 2n)^2} + \frac{1}{(2z + 2n + 1)^2} - \frac{1}{(2z + n)^2} \right] = 0$?

I am reading a textbook, and it uses the equation $\sum_{n = 0}^\infty \frac{1}{(2z + 2n)^2} + \sum_{n = 0}^\infty \frac{1}{(2z + 2n + 1)^2} = \sum_{m = 0}^\infty \frac{1}{(2z + m)^2}$. Could ...
2
votes
1answer
43 views

Convergence of the complex series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{z+n}$

I don't have familiarity with complex series. I know this series won't converge if $z$ is a negative integer. Apart from that, I know that the series does not converge absolutely (because the absolute ...
2
votes
3answers
25 views

Conditionally convergent - limit of series'

Let $\sum_{n=1}^\infty a_n$ be conditionally convergent. Let $k_n:= \max(a_n,0),l_n:=-\min(a_n,0)$ for $n\in \mathbb{N}$ and show that $\sum_{n=1}^\infty k_n =\infty $ and ...
2
votes
1answer
44 views

Ratio of Gamma Functions

Is it possible to show that: \begin{align} \frac{\Gamma\left(\frac{1}{78}\right) \Gamma\left(\frac{29}{78}\right) \Gamma\left(\frac{35}{78}\right) \Gamma\left(\frac{53}{78}\right) ...
1
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0answers
37 views

Can we find the closed-form of the series?

I want to calculate the series $$ F(N,g)=\frac{1}{g^N}\sum_{m=0}^{N(g-1)}\Big(\sum_{i=0}^{[m/g]}(-1)^i\binom{N}{i}\binom{N-1+m-gi}{N-1}\Big)^2 $$ where $g=2,3,4,\cdots$, and $N$ is any positive ...
1
vote
1answer
24 views

Finding the radius of convergence of power series

$$f:(-\rho,\rho)\rightarrow R:x+(\sum_{k=3}^{\infty}\frac{(-1)^k*x^{(2k)}}{2k(2k-1)}$$ I tried using the ratio test but a friend of mine said that it only accounts to even numbers and not the odd ...
0
votes
0answers
14 views

Rearranging series' to converge to a certain point.

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
4
votes
3answers
81 views

Convergence of $\prod\limits_{k=1}^n \left(1+\frac{1}{k^2}\right) $

How can you determine whether the limit $$ \displaystyle \lim_{n\to\infty} \prod_{k=1}^n \left(1+\frac{1}{k^2}\right) $$ exists? Thanks for any hints!
0
votes
1answer
44 views

Convergence of an Integral series of Trig Function.

$$\int^{+\infty}_{1}\left(\tan\left(\frac{1}{x}\right)\right)^2dx$$ I just have to show in a simple comparison with a known derivative if this function converges. My guess is to substitute $1/x$ for ...
0
votes
2answers
63 views

Let $s = 1 + q + q^2 + q^3 + … (|q|<1)$ and $S = 1 + Q + Q^2 + Q^3 + … (|Q|<1)$. Then sum $1 + qQ + q^2Q^2 + q^3Q^3+…$ is equal to?

Can someone give me some hint for this task.I know those two are geometric sequence but I don't know how to even start solving this. Solution is $$\frac{sS}{s + S -1}$$
1
vote
4answers
79 views

Calculate $\sum_{k=1}^\infty \frac{1}{k^2-L^2}$ and $\sum_{k=1}^\infty \frac{1}{\left(k-\frac{1}{2}\right)^2-L^2}$

Calculate $$\sum_{k=1}^\infty \frac{1}{k^2-L^2}, \ \ \ \sum_{k=1}^\infty \frac{1}{\left(k-\frac{1}{2}\right)^2-L^2}$$ for $L<1/4$. The two series is always positive by $L<1/4$ and they ...
2
votes
1answer
19 views

Strict inequality linked with sequence

Suppose a sequence $a_{n}$ is defined in following way: $a_{1}=1, a_{n}=a_{1}a_{2}a_{3}…a_{n-1}+1, n \geq 2$. Prove, that for every natural number $m$ an inequality holds ...
1
vote
0answers
28 views

A sufficient condition for a series of functions to be $\mathcal{C}^i$ (Differentiable)?

Suppose we have the Fourier Series : f(x)=$\sum_{k=1}^{\infty} C_k f_k(x)$=$\sum_{k=1}^{\infty} C_k \sin(kx)$ defined in $(a,b) \in \mathbb{R}$ Using Dirichlet criterion I have shown the sum is ...
0
votes
3answers
32 views

Trouble with the proof of convergence of a series.

$\sum_{n=1}^\infty \frac{n}{(-2)^n}$ I tried using D'Alembert's Ratio on it and this is how far I got: $\frac{(n+1)}{(-2)^{n+1}}\frac{(-2)^n}{n}=\frac{n+1}{(-2)\cdot ...
-1
votes
0answers
26 views

Rearrangement of series' - not absolutely convergent [duplicate]

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
0
votes
1answer
57 views

Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$

Functions in D are finite test functions in $C^\infty(\mathbb{R})$ D' are distributions (genralized functions) Do I have to check that $\forall \phi \in D$: $\lim_{\epsilon \to 0} ...
3
votes
1answer
62 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
0
votes
4answers
73 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
-1
votes
0answers
54 views

Product of divergent and convergent sequence

I have the following question: Let $\{a_n\}$ be a sequence of positive real numbers that converges to $a.$ Find the value for $a$ so that the series ...
2
votes
1answer
37 views

Lebesgue integration

if $f : \mathbb{R} \to \mathbb{R}$ is continuous function which is Lebesgue integrable on $\mathbb{R}$ then show that there is sequence $x_n$ which goes to infinity and $x_n f(x_n)$ goes to $0$. ...
1
vote
2answers
162 views

How was the equation re-written?

This question is a part of inhomogeneous recurrence relations (IHR). The actual question was Find a solution to $a_n - a_{n-1} = 3(n-1)$ where $n \ge 1$ and $a_0 = 2$. While going through the ...
0
votes
1answer
28 views

Snowflake-sequences - Area - Circumference

Consider the following inductively defined snowflakes-sequences: $S_1$ is an equilateral triangle with edge length $l_0$, and $S_{n+1}$ emerges from $S_n$ by dividing each edge by 3 and the middle ...
6
votes
4answers
261 views

When will it diverge? When will it converge?

Test for what $x\in \mathbb{R}$ the series $\sum_{n=0}^\infty nx^n$ converges and for what $x\in \mathbb{R}$ it diverges. Determine the limit of sequence for the case of the convergence. ...
0
votes
0answers
15 views

How can one prove that the Thue-Morse sequence is a morphic sequence?

The Thue-Morse sequence is the infinite sequence $\mathbf{a} = (a_n)_{n \geq 0}$ with $a_n = 0$ if the sum of the digits of the binary expansion of $n$ is even and $a_n = 1$ otherwise. Consider the ...
1
vote
2answers
35 views

Series' - Convergence - Limit of sequences

Examine the following series' for convergence: a)$\sum_{n=1}^\infty \frac{n^3\cdot 3^n}{n!}$, b)$\sum_{n=1}^\infty\frac{n}{(-2n)^n}$, c)$\sum_{n=1}^\infty \frac{n!}{n^n}$, ...
0
votes
2answers
37 views

We draw 5 cards from a standard deck of 52 cards without replacement.

We draw 5 cards from a standard deck of 52 cards without replacement. Find the probability of drawing all the cards of the same suit. I think the answer is P of same suit = ...
1
vote
3answers
41 views

Convergence of the series $\sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right)$ [closed]

Does the series $$ \sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right) $$ converge?
0
votes
1answer
34 views

Ratio test, Root test, and Divergence test related.

(I) Ratio test: If the result is smaller than 1 then the sum is convergent, and if the sum is larger than 1 then the sum is divergent, and that got me thinking if negative infinity (smaller than 1) ...
2
votes
4answers
94 views

How would you prove this converges? $\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$

How would you check if this converges or not? $$\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$$ It looks like a telescopic sequence so I thought I'd first write the beginning values: ...
4
votes
4answers
148 views

How to prove that $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$

I have this sequence: $$\sum_{1}^{\infty} \frac{1}{n^3}$$ and I need to prove that: $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$ So basically I know that this sequence converges using the integral ...
1
vote
1answer
55 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
-1
votes
1answer
97 views

prove the convergence sequence [closed]

How to prove that the sequence is convergent? ${a_{n+2} = {\sqrt{a_{n+1}} + \sqrt{a_{n}}}}$ , where $a_{1}=1$ and $a_{2}$ is a positive number.