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

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0
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1answer
45 views

find the function f(r)

If $$W(r)= \frac{2r+1}{r(r+1)}$$ Express $W$ in the form $$W(r)= f(r) - f(r+1)$$ I tried doing this with partial factors but ended up in getting the + sign instead of - Please help
0
votes
1answer
27 views

Evaluate supremum of sequence $\frac{n(n+1)-2\left\lfloor \frac{n}{2}\right\rfloor \left\lceil \frac{n}{2}\right\rceil}{n(n+1)}$. [closed]

Find the supremum of sequence $$\frac{n(n+1)-2\left\lfloor \frac{n}{2}\right\rfloor \left\lceil \frac{n}{2}\right\rceil}{n(n+1)}$$ where $n\in\mathbb N$ and $\lfloor x\rfloor$ is the largest integer ...
1
vote
1answer
81 views

Some issues with proving that a sequence is convergent

I recently tried (in the sense that I believe the thesis holds) to prove that, given $a\in\mathbb{R}^+$, there exists $$\lim_{n\rightarrow+\infty}\sqrt[n]{\sum_{k=0}^{\lfloor n/5\rfloor}{n-4k\choose ...
1
vote
1answer
26 views

Limit of a sequence = subsequence of a limit subsequence

Let there be a sequence $a_n$ and for all subsequence of it $b_n$ there is a subsequence $c_n$ that convergence to $a$ Prove: $a_n \rightarrow a$ Where should I start?
1
vote
2answers
62 views

Divergent, convergent series

Let $p$, $q \in \mathbb{R}$ and see the series $$ \sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q} $$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and ...
0
votes
1answer
70 views

Sum of the series $\sum_{n\geq 1}\frac{n^k}{n!}$, aka Dobinski's formula [closed]

What is the sum of the following series? $k$ is a natural number. $\sum_{n\geq 1}\frac{n^k}{n!}$ It appears to always provide an integer multiple of $e$.
1
vote
1answer
46 views

Poisson power series

We have a Poisson power series of $$Y=\sum\limits_{k=0}^{\infty}e^{-\pi\lambda v^2}\frac{(\pi\lambda v^2)^k}{k!}(A)^k $$ If we have a disk with radius $v$ where A is defined as the density of a ...
1
vote
2answers
35 views

The “Smooth Factor” in a number sequence

I'm trying to figure out a programming problem that has mathematical foundations. The problem says: For an array $$a = a_1, a_2, ..., a_n$$ of values, the smooth factor of $$a$$ is the length of a ...
0
votes
1answer
48 views

Determine the Taylor Series for $(1+x)^n$ about $x=0$

Having trouble solving this. I get to expanding to this: $$1^n + n(1^{n-1})\cdot\frac {x!}{1!}+n(n-1)\cdot 1^{n-2} \cdot \frac {x^2}{2!} +n(n-1)(n-2)\cdot 1^{n-3}\cdot \frac {x^3}{3!}\dots$$ Where do ...
2
votes
2answers
47 views

How to prove this definition of the Riemann Zeta Function?

I have to prove that $\sum_{n=1}^{\infty} \frac{1}{n^s}$ = $\frac{1}{1-2^{1-s}} \sum_{n=1}^{\infty} \frac{-1^{n+1}}{n^s} $ I know I am supposed to show my work, but I have no clue on how to ...
0
votes
1answer
50 views

Where am I going wrong? When I try to go farther than this they stop being equal to each other.

If $$5+8+11+ \dots +(3n+2)=\cfrac{n(3n+7)}{2}\;,$$ then $$5+8+11+ \dots +(3n+2)+\big(2(3n+1)+2\big)=\cfrac{(n+1)(3n+8)}{2}\;.$$
0
votes
2answers
24 views

representing an iteration loop in math

I have a computation step where $$ a_{n+1} = f_1(a_n) $$ That is, $a$ at step $n+1$ is some function $f_1$ of $a$ at step $n$. I want to iterate till an $N$ where $f_2(a_N) = b$ (where $f_2$ is ...
1
vote
0answers
44 views

Non-uniform convergence example

For the past couple of days, I've been trying to come up with a example for a problem in which $\sum_{k=1}^{\infty} |f_{k}(x)|$ does not converge uniformly but $\sum_{k=1}^{\infty} |f_{k}(x)|$ ...
0
votes
0answers
12 views

Function whose fixed points are a convergent sequence with derivatives at each term $\neq 1$ and derivative at limit point $=1$

The question asks to find an example of a function where the fixed points of the function are a sequence that converges to a fixed point, where the derivative of the function at each of the fixed ...
3
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0answers
35 views

Dealing with non-constant term in Binomial Theorem question

I am wondering this. Suppose I have a sequence $\{\varepsilon_n\}_{n=0}^\infty$ and elements of this sequence are part of a binomial type expression: For example, my expression is ...
1
vote
2answers
30 views

A sequence of distributions converges to a certain distribution.

Given the sequence of functions: \begin{equation} f_n(x)=tanh(nx) \end{equation} and knowing that: \begin{equation} \lim_{x \to \pm \infty}f_n(x)=f(x)=\begin{cases} -1, & x<0 \\ 1, & ...
0
votes
3answers
25 views

Linear recursive sequence in closed-form function

I've been trying to find an answer for a question for some time, and I've done some Google searching but can't seem to figure out exactly how to solve it. It is a linear recursive sequence, and it ...
0
votes
2answers
25 views

Summation rule index

In the simplification shown below, The index are changed but when i follow the index change rule, i cant find the same final answer. $$\sum^{n-1}_{i=0} \frac{n}{n-i}=n\sum^{n}_{i=1} \frac{1}{i}$$ ...
0
votes
2answers
29 views

Radius of convergence for series

I was looking at an example for finding the interval of convergence. $$F(x)=\sum_{n=1}^{\infty}\frac{(x-3)^n}{n}$$ Use the ratio test. $$ \lim_{n\rightarrow\infty}\left | ...
2
votes
2answers
48 views

Is this true: $\sum_{n=1}^\infty a_n = \sum_{n=2}^\infty a_{n-1}$ ??

Does this equality hold? $$\sum_{n=1}^\infty a_n = \sum_{n=2}^\infty a_{n-1}$$
2
votes
2answers
51 views

Calculating limit of a series

So I have the following series: $$\sum_{2n}^{4n} \frac 1k$$ and I need to find the limit of it when n appraoches infinity: $$\lim_{n\to\infty}\sum_{2n}^{4n} \frac 1k$$ I tried to consider this sum as ...
0
votes
1answer
20 views

solving limit of sequence by sandwich theorem

I am trying to find the limit (as n tends to infinity) of $$a_n=\left(3^n+9^n\right)^{1/n} $$ and $$b_n=\left(1+ \frac{7}{8n^3}\right)^{n^3}$$ What I am considering is to use Sandwich Theorem ...
0
votes
1answer
13 views

Existence of limit of certain numerical sequence

suppose $(a_n)_{n=1}^{\infty}$ is a sequence satisfying the following: (1)$(a_n)_{n=1}^{\infty}$ is decreasing and $a_n\geq 0$ (2)$\displaystyle\lim_{n\to\infty}a_n=0$ (3)for some positive ...
0
votes
5answers
54 views

How can I find the general formula for the following real sequence

How can I find the general formula for the following real sequence $$(x_n)_{n \ge0}=(1,0,-1,0,\frac{1}{2},0,\frac{-1}{6},0,\frac{1}{24},0,\frac{-1}{120},\ldots)$$ I just know $x_0$ to $x_{10}$ so how ...
1
vote
1answer
30 views

convergence/divergence of a series power of n vs factorial

prove convergence/divergence of: $$\sum\limits_{v=1}^\infty (-1)^n\cdot \frac{3^{n^2}}{(n!)^3}$$ we should first check if $$\sum\limits_{v=1}^n \frac{3^{n^2}}{(n!)^3}\rightarrow 0$$ using the ...
0
votes
1answer
33 views

Convergence of $\sum_ {n=1}^\infty (1-n\sin\frac1n)^\alpha$ and $\sum_ {n=1}^\infty 2^n (\tan x)^{n^2}$

I was trying to solve a question of an entrance exam. I am having trouble in a particular type of problems. Please help me to solve. (Actually my last 2 questions are also from these exam papers. I ...
0
votes
1answer
65 views

How can I solve the following differential equation by power series [closed]

How can I solve the following differential equation : $$zw''+(1+4z^2)w'+4z(1+z^2)w=0 $$by power series near the regular singular point $z=0$
1
vote
2answers
26 views

A permutation and combination question (plus summation of a series)

A student is browsing in a second-hand bookshop and finds $n$ books of interest. The shop has $m$ copies of each of these $n$ books. Assuming he never wants duplicate copies of any book, and that he ...
5
votes
1answer
45 views

converge/divergence of a series with a n-th root

for the following series, prove for convergence/divergence $$\sum\limits_{v=1}^\infty \frac{\sqrt[n]{n}}{\sqrt[n]{n!}}$$ $$\sum\limits_{v=1}^\infty ...
0
votes
0answers
35 views

Taylor Series Formulae

How are the two following forms of the Taylor expansion equivalent? The one I've learnt is $$f(x+h)=f(x)+hf'(x)+\frac{h^2}{2!}f''(x)+...$$ But I've now come across the version $$ ...
1
vote
1answer
30 views

Bounded sequence

I'm currently learning for my math exam and I am stuck at the following problem: If a real sequence $a_n$ has an upper and a lower boundary $s_u$ and $s_\ell$, then $s=\max\{s_u,s_\ell\}$ is a ...
0
votes
2answers
33 views

Sum to infinity of geometric sequences (simultaneous equation)

I tried to eliminate a: $76-76r=a$ $36-36r^3=a$ I ended up with $40-76r+36r^3=0$ I was unsure on how to proceed so I checked the mark scheme and I should have ended up with a quadratic ...
0
votes
0answers
33 views

convergence of a sequence of iid random variable [closed]

I am given a sequence $\{ c^k X_k\}_{k\ge0}$ where $c>0$, $X_k$ are iid random variables with finite variance. For what values of $c$ does this sequence converges almost surely?
1
vote
2answers
51 views

Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$

$\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|>1$. There are two facts that my professor uses that I am confused about. The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any ...
-3
votes
2answers
47 views

Convergence criteria for the series $1+2x+4+6+8+…$ [closed]

I need help for the sum of the following series and the values of $x$ for which the series converges? $1+2x+4+6+8+…$ Any help would be appreciated. Thank you.
2
votes
6answers
43 views

Show that $\sum_{n=0}^{\infty}xe^{-n^2x}$ converges pointwise on $(0,\infty)$

$\sum\limits_{n=0}^{\infty}xe^{-n^2x} = x\sum\limits_{n=0}^{\infty}e^{-n^2x}$. Apparently the series converges pointwise on $(0,\infty)$ by a limit comparison test, but I cannot see what series I ...
0
votes
1answer
41 views

Suppose that Σ (a_n)*(x^n) has a finite radius of convergence R and a_n>=0 for all n.

Show that if the series converges at $R$,then it also converges at $-R$. What I have done is, since the given power series converges at $R$ (finite quantity), then by $n$-th term test $\lim a_nR^n=0$. ...
7
votes
1answer
44 views

Sequences of real numbers, arithmetic mean.

Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after ...
-1
votes
1answer
33 views

How to find value of i when ∑ from k=1 to i is defined by a recursive formula and equals 982?

Thanks for the pointers! Here's updated and edited question I'm trying to find the number of days it takes to reach 982 miles when you start traveling at 18 miles/day and decrease your speed by 2% ...
0
votes
1answer
46 views

Question about the series $\sum\limits_{n=0}^{\infty} e^{-nx^2}$ on $(0,1)$

Consider the series $\sum\limits_{n=0}^{\infty} e^{-nx^2}$ on $E = (0,1)$. If we allowed $x=0$, what would happen to the series? On one hand, we could write it as $\sum\limits_{n=0}^{\infty} ...
1
vote
2answers
40 views

Is there an easy way to calculate $(1-n/4)(1-n/5)…(1-n/30)$ for any integer $n$?

Is there an easy way to calculate $(1-n/4)(1-n/5)....(1-n/30)$ for any integer $n$ .I tried to expand the bracket, but it seems a lot of troubles.
1
vote
1answer
23 views

Big O properties

I'm studying some properties about "Big O". I begin to understand the idea behind of this but have some trouble with the following equality: $$O(1/n)=O(1/n^{2})$$ I don't understand it. I would like ...
0
votes
2answers
23 views

metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - ...
15
votes
0answers
165 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...
0
votes
0answers
42 views

How fast things con/diverge. [closed]

Is there some kind of standard listing for how fast common series converge/ diverge? For instance, is it true that this order holds (diverges faster closer to the top)? Pretend there's a ...
1
vote
1answer
32 views

Simple calculus series question; convergence of $\sum_{j = -\infty}^{\infty} \frac{1}{z - j}$

So I am having a brain fart and I cannot rigorous write (in my head) why the series in the question does not converge. I know it has to do with the harmonic series. Is it because if $|z| \to \infty$, ...
0
votes
0answers
20 views

Why would you choose the Method of Frobenius over a Power Series solution to solve a DE?

I'm trying to determine where it would be more appropriate to use one or the other. To further clarify: Where would it make more sense to use: $y=\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}$ instead of ...
1
vote
2answers
35 views

limit of sequnce in real numbers

For every tow real numbers $a$ and $b$,with this condition that $0<a<b$ ,define sequence ${x_{n}}$ to the following: $x_{1}=a$ $x_{2}=b$ ${x_{n}}=\frac{ x_{n-1}+ x_{n-2} }{2}$ (for ...
0
votes
0answers
21 views

Series of Riemann Zeta values analytic continuation.

What is the value of the series?: $$\sum_{k=1}^\infty\frac {\zeta(-k)} k$$ Where $\zeta(z)$ is the Riemann Zeta function and for every negative integer $n$ we have $\zeta(n)=-\frac {B_{n+1}} {n+1}$. ...
2
votes
2answers
33 views

Convergence of an infinite logarithmic series.

$$\sum_{n=1}^\infty\ln\left(1+\frac{1}{n^a}\right)$$ Depending on the parameter $a\gt 0$, test the convergence. What method should I use?