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

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2
votes
1answer
44 views

For $0≤a≤b≤c$, show that $\lim\sqrt[n]{a^n+b^n+c^n}=c$

For $0≤a≤b≤c$, show that $\lim\sqrt[n]{a^n+b^n+c^n}=c$ I think I am making some silly mistake with my "proof". If it is indeed correct, another question emerges. So, my attempt is: Since $0≤a≤b≤c$, ...
0
votes
0answers
17 views

Power series - interval of convergence

For $f(x) = \sum_{n=2}^{\infty} \frac{(x+1)^n}{n(n-1)}$ I have showed that $f'(x) = \sum_{n=1}^{\infty} \frac{(x+1)^n}{n}$ and that $f''(x)=\frac{-1}{x}$ at all points where f converges absolutely. ...
3
votes
3answers
45 views

Prove sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$

I am asked to verify that the sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$: $$\lim \frac{1}{6n^2+1}=0.$$ Here is my work: $$\left|\frac{1}{6n^2+1}-0\right|<\epsilon$$ ...
3
votes
1answer
18 views

Fixed points and infinite series

Consider the formula $1 + \frac{y}2$. This has a fixed point at $y = 2$. And if we use the equation $y = 1 + \frac{y}2$ to substitute for $y$ in our formula, we get $1 + \frac{1 + \frac{y}2}2$, or $1 ...
5
votes
0answers
45 views

Disprove, fix, prove: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing

Prove the statement wrong, fix it, then prove the new statement: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing I think I'm headed in the right direction with this but I'm ...
2
votes
2answers
32 views

Prove bounds of a strictly increasing sequence using integrals to approximate

For the strictly increasing sequence $ \ x_n = \frac1{1^2} + \frac1{2^2} + \frac1{3^2} +\cdots+\frac1{n^2},$ for $n\ge1$. (a) Prove the sequence is bounded above by $2$; deduce that is has a limit ...
0
votes
1answer
15 views

Power Series - differentiation and absolute convergence

I am having problems with the following exercise: Ex. 1. Let $f(x) = \sum_{n=1}^{\infty} \frac{(x-1)^n}{n}$ (i) Find the convergence interval. Here I let $f(x) = \sum_{n=1}^{\infty} ...
7
votes
2answers
217 views

Infinite sum of reciprocals of pentagonal numbers

How do I find this sum: $$\sum_{n=1}^\infty \frac{1}{p(n)}$$ where $p(n)=\dfrac{n(3n-1)}{2}$ is the $n$th pentagonal number? I know it is a convergent series, but I don't know if the sum can be ...
3
votes
1answer
48 views

Showing that a recursively defined sequence is decreasing.

A colleague of mine is interested in finding out how to show the following: Prove that the sequence $(a_n)$ defined by ...
1
vote
1answer
18 views

Converging subsequences and subsets having infinite elements

We have a metric space (V,d). Proof that the following two properties are equivalent. a) Every sequence $a_n \in V$ has a subsequence which converges to a element $x \in V$ b) For every ...
2
votes
0answers
33 views

How to solve Recurring Series problem [on hold]

How do you solve recurring series problem? $$2-x+5x^2-7x^3+...$$ My work: The scale of relation is $1+x-2x^2$ and Sum=$\frac{2+x}{1+x-2x^2}-\left(\frac{\text{some expression}}{1+x-2x^2}\right)$ ...
0
votes
1answer
19 views

Sum of elements in a sequence

Let $a_n$ be a sequence in $\mathbb{R}$ and $a\in\mathbb{R}$. Suppose that $N \in \mathbb{N}$, $\epsilon >0$ and for every $n > N$ $|a_n -a|<\epsilon$. Show that for every $n>N$ the ...
2
votes
3answers
27 views

Find all the values of x, for which the series converges.

$\sum\limits_{n=1}^∞ (x^2/(x^2+4))^n$ I did try to use the ratio test and I ended up with $| x^2/(x^2+4)|<1$ I don't have any idea what to do after this, how do I solve for x?
1
vote
1answer
13 views

Help with Proving : Period estimation for for concatenated sequences

Assuming I have two 8-bit random number sequences $s[n]$ and $d[n]$ which each have a period of $X$ and $Y$ respectively. Therefore: $$s[n+X] = s[n]\\ d[n+Y] = d[n]$$ If they were concatenated ...
3
votes
1answer
43 views

Is there a total summation function?

I define a summation function to be a partial function $F$ from infinite sequences of real numbers to the extended reals, such that: (1) Sequences which are zero in all but possibly one position are ...
1
vote
1answer
55 views

Help with the proof that the sum of all the roots of a complex number is zero

If a complex number $z \neq 0$ has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ {i (\theta +2\pi j) }/{n} } $$ For $j=0,1,2,...,n-1$ Thus, the summation of all the roots ...
-7
votes
0answers
39 views

How can I prove or disprove that every series converges to philosophy? [on hold]

Let $x_0$ be any wikipedia page. Let $x_n$ be the page that the first non-etymological link on $x_{n-1}$ leads to. How can I prove that $x_n$ eventually converges to philosophy? Technically it seems ...
1
vote
1answer
29 views

Solving this finite factorial/binomial series

I am staring at the following finite series: $$ \sum_{k=1}^s A_k Y^k $$ I am trying to solve it for $Y$. $A_k$ is given by $$ A_k = {s \choose k}\nu^k (1-\nu)^{s-k}$$ I already solved that $$ ...
1
vote
3answers
47 views

A recursive sequence is defined by…

A sequence is defined recursively by $a_1=1$ and $a_{n+1} = 1 + \frac{1}{1+a_{n}}$. Find the first eight terms of the sequence $a_n$. What do you notice about the odd terms and the even terms? By ...
1
vote
1answer
21 views

What are the basic rules for manipulating diverging infinite series?

This is something that I played around with in Calc II, and it really confuses me: $s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots = \infty$ $s - s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots $ $ \ \ \ \ ...
1
vote
0answers
61 views

An identity involving the Bessel function of the first kind $J_0$

Today, computing $\int_{0}^{\pi/2}\sin^2(\sin^2 x)\,dx$, I found an interesting identity: $$\sum_{k\geq 0}\frac{(-1)^k(4k)!}{(2k)!^3 4^k}=\cos(1)\cdot\sum_{k\geq 0}\frac{(-1)^k}{k!^2 4^k}.$$ How would ...
4
votes
0answers
44 views

How many possible shuffles can be won perfectly?

It is known that the possible shuffles of a deck of cards is $52!$, or ~$80658175170943878571660636856403766975289505440883277824000000000000$ different combinations. I have become aware of a game ...
1
vote
3answers
64 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $x_n\to 0$, such that $f(x_n)=0$, for all $n$. Then $f\equiv 0$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
2
votes
2answers
39 views

How does this manipulation of summations work?

I am reading some mathematics in which is the following algebraic manipulation. $$ \begin{align} \exp(x)\exp(y) & = \left(\sum_{n = 0}^\infty \frac{x^n}{n!}\right) \left(\sum_{m = 0}^\infty ...
2
votes
2answers
44 views

If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges?

If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges? I ...
2
votes
2answers
33 views

Determine the set of values of $x$ such that this series converge

Determine the set of values of $x$ such that this series converge: $$\sum^{\infty}_{n=1} \frac{e^n+1}{e^{2n}+n} x^n$$ My work: If $x\geq e$, we have $$\frac{e^n+1}{e^{2n}+n} x^n \geq ...
0
votes
1answer
37 views

Differentiate this power series

I am working on a problem which involves the differentiation of a power series. I know that that the following holds. Let $R$ be the radius of convergence of the power series $\sum_{n = 0}^\infty ...
2
votes
2answers
28 views

Proof that repeated sum equals binomial formula

Let $s, d$ be positive integers. Can you prove the following general formula for the repeated sum? I developed this problem on my own, but is it a well known result? $$\sum_{i_1 = 0}^s \sum_{i_2 = ...
0
votes
0answers
36 views

Doubt on the comparison test: can I still evaluate $\lim \limits_{n \to \infty} \frac{a_{n}}{b_{n}}$ if $b_{n}$ might be $0$ for some $n$?

Suppose that $(b_{n})_{n \in \mathbb{N}}$ is a sequence which is not identically equal to $0$ (but which may have elements equal to $0$). Suppose also that I know that $\sum b_{n}$ converges ...
0
votes
0answers
25 views

Showing sequence function is monotone [duplicate]

Is this sequence of functions monotone? $$f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}, \forall x \in[0,+\infty)$$ Where $\varphi:[0,+\infty)\to \mathbb{R}$, $1/2\leq \varphi(x) <1$ ...
2
votes
3answers
47 views

Is the argument I used to evaluate the convergence of the series $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$ right?

If $a,b,c$ be real constants, analyze the convergence of $$\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$$ What I tried to: I compared the general term of my series to $\frac{1}{n}$: ...
2
votes
0answers
13 views

Special $\omega(n)$-sequence

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$. The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a ...
2
votes
7answers
86 views

Tell if a sum is convergent $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$

$$\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$$ I tried to solve this by saying that $$\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}$$ I then made two sums like this: $$\sum\limits_{n=1}^\infty ...
0
votes
2answers
35 views

Calcultaing function limit of a limit sequence

How do I even start? $$\lim_{x \to 1^+} \lim_{n \to \infty}{x^n \over x^n + 7}$$ I see that it should be $1$ but how do i prove it?
3
votes
1answer
40 views

Asymptotic ratio of two series

Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow ...
-3
votes
0answers
24 views

How to quickly find $2^x \pmod N$ cycle

$N$ is a large composite number. If we don't have the factorization for $N$,how do we quickly find the order of $2$ in $(\mathbb{Z}_N, \cdot_N)$? Example. Suppose $N$ is ...
3
votes
1answer
33 views

Showing a sum of $\vert f(x+k)\vert $ belongs to $L^{\infty}$ if $f,f'\in L^1$

I am working on this Suppose that $f,f'\in L^1(\mathbb{R})$. Then $\sum_{k= 0} ^{\infty}\vert f(x+k)\vert\in L^{\infty}([a,b])$ for any $a,b\in \mathbb{R}.$ Idea: Let $i$ be any integer. $\int_i ...
1
vote
1answer
36 views

Series in hyperbolic sines.

I was looking into a problem and I arrived to something in which I want to expand some function $\varphi(x)$ in series of hyperbolic sines, something like: ...
-1
votes
1answer
26 views

Uniform convergence of series $\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $

I'm having trouble with uniform convergence. I need to prove that $$\sum_{n=0}^\infty \left(\frac{z-1}{z+1}\right)^n $$ converges locally uniformly in the half-plane $Re z >0$ and find its sum. ...
2
votes
1answer
78 views

$1-1+1-1+1-1+\cdots = \frac 12$ proof?

Note: I claim no credit for this "proof". A friend came up with it and I thought it was pretty cool. Let's say you wanted to prove that $$1-1+1-1+1-1+ \cdots = \frac 12$$ Well, as mentioned before, ...
-2
votes
1answer
27 views

Determine the sum of the following series. [on hold]

Determine the sum of the following series: $$\sum_{n=1}^\infty\frac{(-3)^{n-1}}{n^5}$$
0
votes
5answers
55 views

Define a sequence {$\ x_n$} recursively, show it is strictly decreasing

Define a sequence {$\ x_n$} recursively by $$ x_{n+1} = \sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1 $$ Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.
1
vote
4answers
29 views

Find the sequence of partial sums for the series $a_n = (-1)^n$ Does this series converge?

Find the sequence of partial sums for the series $$ \sum_{n=0}^\infty (-1)^n = 1 -1 + 1 -1 + 1 - \cdots$$ Does this series converge ? My answer is that the sequence $= 0.5 + 0.5(-1)^n$. This makes a ...
0
votes
1answer
21 views

Calculate the limit of the sequence by applying the limit laws?

I'm not sure how to approach this problem since its a bit different to the usual questions about calculating limits .
-5
votes
3answers
40 views

Classification of Series $(-1)^n$ [on hold]

Does it converge or diverge or we can't tell? $$∑_{n=1}^{\infty}(-1)^n$$ Or is there simply no concrete answer? Thanks in advance.
2
votes
1answer
41 views

What is the Taylor series of $e^x$ centred at $3$?

$$ \sum_{k=0}^n \frac{e^3}{n!}(x-3)^n $$ This is my answer - is it correct?
2
votes
0answers
31 views

How to find coefficients in a power of a trig series?

Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: ...
4
votes
2answers
386 views

Does this sum converge, is my solution good?

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7}+1)^{1/2}} $$ I would say that it doesn't converge, cause I would write this as: $$ $$ $$ \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ when $$ \lim_{n\to ...
-1
votes
0answers
9 views

Problems understanding Autocorrelation/Autokovariance

I am having some trouble understanding the concept of autokovariance/autokorrelation with a timelag l. From how i understand it, it is the kovariance/korellation a series has, with a timelagged ...
0
votes
1answer
18 views

Bernoulli odd numbers are 0 $B_{2n+1}=0,\;n>0$

I left the Maclaurin expansion of the function $f(x)=x/(e^x-1)$ $$\frac{x}{e^x-1}=\sum_{k=1}^{\infty} B_k\frac{x^k}{k!}=B_0\frac{x^0}{0!}+B_1\frac{x^1}{1!}+\sum_{k=2}^{\infty}B_k\frac{x^k}{k!}$$ $$ ...