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

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0answers
19 views

Proof that Newton expansion over derivatives has the properties of an integral

Let's consider a Newton expansion over consecutive derivatives of a function: $$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ Can it be proven that such ...
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1answer
501 views

Show that $\sqrt{2+\sqrt{2+\sqrt{2…}}}$ converges to 2

Consider the sequence defined by $a_1 = \sqrt{2}$, $a_2 = \sqrt{2 + \sqrt{2}}$, so that in general, $a_n = \sqrt{2 + a_{n - 1}}$ for $n > 1$. I know 2 is an upper bound of this sequence (I proved ...
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1answer
28 views

Prove if $\{t_n\}_{n\in\mathbb{N}}\to t$ and $t_n\geq 0\forall n\in\mathbb{N}$, then $t\geq 0$

I've been working on some sequence practice problems in Steven Lay's Introduction to Analysis With an Introduction to Proof for my introductory real analysis course, as we are starting our unit on ...
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1answer
85 views

Example of two series with certain properties?

Find 2 series $\sum a_k$ and $\sum b_k$ such that $\sum b_k$ converges conditionally, $\dfrac{a_k}{b_k} \rightarrow 1$ as $k \rightarrow \infty$, and $\sum a_k$ diverges. Can someone give me a hint ...
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0answers
42 views

Calculate the Maclaurin series, using binomial series

Calculate the Maclaurin series for the following function This is a note from my teacher through email "Question 5a (this question) is now a bonus question, since it requires binomial series that we ...
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3answers
626 views

proving that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
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2answers
36 views

Integer Part of sequence convergence

I was trying to solve the following exercise. If $(a_n) \in \mathbb{R}$ and $(a_n)\rightarrow {1}/{2}$ show that $[a_n] \rightarrow 0$ , where $[~]$ the integer part. I was trying to solve it using ...
2
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0answers
25 views

Does the series $\sum(-1)^k\binom{x}{k}$ converge for -1<x<0? [closed]

Does the series $\sum_{k=0}^{\infty}(-1)^k\binom{x}{k}$ converge for $-1<x<0$?
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2answers
108 views

Why is the sequence $ a_n = \left(1+\frac{1}{n}\right)^n $ Cauchy?

I was looking at the post: Cauchy Sequence that Does Not Converge And the top answer was this sequence: $ a_n = \left(1+\frac{1}{n}\right)^n$. I understand that this sequence converges to $e$, ...
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1answer
65 views

Is there an easy formula for this sequence?

It is the sequence which represents the maximum number of cycles in an undirected graph with n nodes, n>=3. These graphs have all nodes connected to every other node. How would I count the number of ...
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1answer
75 views

Probabilities for $1$-in-$n$ events over $n$ trials

I know there are lots of related questions on here, but I can't seem to find what I'm looking for. Given some event with, say, a $1$ in $1,000,000$ probability (e.g., $7$ being chosen randomly as a ...
2
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2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
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1answer
127 views

Sum of self power

Is there a formula to calculate the sum of a number to the power of this same number, like: $$1^1 + 2^2 + 3^3 + 4^4 + 5^5 + ... + n^n$$? or $$x^x + (x+1)^{(x+1)} + (x+2)^{(x+2)} + ... + ...
2
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0answers
69 views

What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
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1answer
122 views

$\sum\frac{k^{k/2}}{k!}$ converge or diverge?

Does the following series converge or diverge? $\sum\frac{k^{k/2}}{k!}$ I did the ratio test $\frac{a_{n+1}}{a_n}$ I did $\frac{(k+1)^{(k+1)/2}}{(k+1)!}$ $\frac{k!}{k^{k/2}}$ Then I simplified ...
2
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1answer
39 views

Proof regarding series

I've encountered a recommended practice proof that I'd like some assistance in starting. Suppose that $\sum_{i=1}^\infty an$ and $\sum_{i=1}^\infty bn$ are both series with all positive terms and ...
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2answers
60 views

proof convergence of $(\sqrt[n]{25})_{n\in\mathbb N}$ and $(\frac{2^n}{n!})_{n\in\mathbb N}$

I want to show that $(\sqrt[n]{25})_{n\in\mathbb N}$ and $(\frac{2^n}{n!})_{n\in\mathbb N}$ are convergent. So for the first one I did the following:\begin{align} 25&=(1+\delta_n)^n \\ ...
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1answer
72 views

How to prove that $\,\,\sum_{n=1}^{\infty}\frac{1}{n^{1+i}}\,$ diverges

I am stuck on the following question : Prove that $$\sum_{n=1}^{\infty}\frac{1}{n^{1+i}}$$ diverges, where $i=\sqrt{-1}$ I am not sure how to progress with it. Can someone explain? Thanks and ...
4
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1answer
341 views

Prove that a given recursion sequence converges

I'm given: $$\begin{align*} x_1&=\frac32\\\\ x_{n+1}&=\frac3{4-x_n} \end{align*}$$ How do I go about to formally prove the sequence converges and show it? Thanks in advance.
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1answer
67 views

Proving that $\sum_{n=1}^{\infty}\frac{H_n}{q^n}=\frac{q}{q-1}\log(\frac{q}{q-1})$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{q^n}=\frac{q}{q-1}\log(\frac{q}{q-1})$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$, $|q|>1$).
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1answer
68 views

Sequence Convergence and Limits

Here is a problem I've been working on. I am stuck and wondered if you guys could shed any light. Let $a>0$ and $u_{1}>a$. Consider the sequence $(u_{n})_{n=1}^{\infty }$ defined by: $$ ...
15
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4answers
737 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
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1answer
30 views

Representing Log(z) as a series about z=1 from Log'(z)

I am doing a homework quesiton at the moment and have a question which says to find that a series exapansion about z=1 for Log(z) So far I have this Since $Log'(z)$ = $\frac{1}{z}$ we can turn this ...
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2answers
164 views

Limit of $(1+5/n+6/n^2)^n$ when $n$ goes to infinity [closed]

Find $$\lim_{n \to \infty} \left(1+\frac{5}{n}+\frac{6}{n^2}\right)^n$$
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1answer
20 views

Is that possible for an array to be divergent sequence but convergent series?

Is there possible to construct an array such that when it is consider as a sequence, it diverges. But as series, it converges ??
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1answer
50 views

Proof of divergence of a series

I'd really appreciate some help with this question on my recent math assignment: Show that if $a_n > 0$ and $\lim_{n\to \infty} na_n = L$, where $0 < L < \infty$, then $a_n$ is divergent. ...
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3answers
68 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
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2answers
61 views

Sequence and convergence of subsequences

Suppose that $(a_n)$ is a sequence. Assume that both $(a_{2n})$ and $(a_{2n+1})$ converge to the same $L$. Prove carefully that $(a_n)$ also converges to $L$ I was thinking that $(a_{2n})$ and ...
0
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1answer
25 views

Inverse theorem on product of two convergent sequences

Suppose I have two sequences, $a_n$ and $b_n$. I know that: $\lim_{n\to\infty} a_n=1$ and that $\lim_{n\to\infty} a_nb_n=c$. Does this mean that $\lim_{n\to\infty} b_n$ converges? If so, by ...
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1answer
53 views

How to solve recursive function

I've recently been doing some limits with circuits and such, and I came up with the following equation, $R$ being a constant: $$f(x) = \frac{f(x-1)*R}{R+f(x-1)}+R$$ with $f(1)=2$. I know that this ...
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1answer
44 views

How to construct longitudinal from transversal waves and vice versa?

The above construction of a longitudinal wave out of a transversal wave has been encountered somewhere in an old physics textbook. There are several drawbacks with this construction. The maximum ...
0
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1answer
157 views

The meaning of almost surely convergence

Consider the space of sequences of tosses of a fair coin. In particular let $X_{nH}$ be the number of times that a coin lands on heads in a sequence of $n$ coin flips. Consider statement $S$ below. ...
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0answers
65 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
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3answers
49 views

problem with pattern of sequance

there is a sequence $a_1=1^2, a_2=3^2, a_3=6^2, ...$ I'm thinking if there is a pattern of this progression, but so far I haven't find out. I noticed that the difference of only the square number ...
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1answer
39 views

how prove there are infinite numbers $\frac{a_{i}}{1234}\in N$ and $a_{n+2}=a^2_{n+1}-a_{n}$

let $a_{1}=287,a_{2}=39$,and $$a_{n+2}=a^2_{n+1}-a_{n}$$ show that: this sequence $\{a_{n}\}$ contains infinitely many $a_{i}$,such that $\dfrac{a_{i}}{1234}\in N$ My try: since ...
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2answers
274 views

Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?

Let the sequence $\{a_n\}$ be defined as $a_1 = \sqrt 2$ and $a_{n+1} = \sqrt {2+a_n}$. Show that $a_n \le$ 2 for all $n$, $a_n$ is monotone increasing, and find the limit of $a_n$. I've ...
2
votes
2answers
107 views

Uniform (but not normal) convergence of a series of function

consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$. It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal ...
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1answer
71 views

If the series $\sum_{k=0}^{\infty} a_k$ converges then $a_k$ converges to 0. [duplicate]

If the series $\sum_{k=0}^{\infty} a_k$ converges then $a_k$ converges to 0. How to prove this theorem?
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3answers
326 views

Can any continuous function be represented as an infinite polynomial?

Can any continuous function be represented as an infinite polynomial? Motivation: the antiderivative $ \int^\ e^{-x^2}dx\ $ can be expressed as an infinite polynomial(write Taylor series for ...
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votes
5answers
103 views

Which sequence will be longer? sequence of natural numbers or even numbers [closed]

Its not that I don't know answer for this question, but this question is definitely debatable. the main reason for asking this question is that someone will be able to convince me on the answer as I ...
0
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1answer
39 views

Show that: $\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$

How can I show that: $\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$ I'm struggling to figure out how to start on this question. Should I sum the series and then differentiate it and also ...
2
votes
1answer
54 views

Finding the value of $a^x$

We have the series expansion $e^x = 1 + \frac{x}{1!}+ \frac{x^2}{2!}+ \frac{x^3}{3!}+ \frac{x^4}{4!}...\infty$. Is it possible to write $a^x$ in the similar form, where ...
5
votes
2answers
175 views

Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$

How can I find the value of the following sum? $$\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$$ $F_n$ is the Fibonacci number.($F_1=F_2=1$)
5
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2answers
864 views

Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$

Is it true that for $x\in[0,2\pi]$ we have $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$$ How can I prove it? For other intervals what is the value of above ...
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1answer
78 views

A problem on sequence and series

Suppose we have a decreasing sequence $\{x_n\}$ which converges to $0$. Then is it true that the sum $$\sum_{n=1}^{\infty}\frac{x_n-x_{n-1}}{x_n}$$ diverges ?
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2answers
83 views

Evaluation of a series (possibly related to Binomial Theorem)

I have the following series: $$1 + \frac{2}{3}\cdot\frac{1}{2} + \frac{2\cdot5}{3\cdot6}\cdot\frac{1}{2^2} + \frac{2\cdot5\cdot8}{3\cdot6\cdot9}\cdot\frac{1}{2^3} + \ldots$$ I have to find the ...
5
votes
4answers
206 views

Why are these equations equal?

I have racked my brain to death trying to understand how these two equations are equal: $$\frac{1}{1-q} = 1 + q + q^2 + q^3 + \cdots$$ as found in ...
0
votes
2answers
66 views

Determine the value for which a sequence is an arithmetic progression.

We have the following sequence $$ -a, -\dfrac{a}{b}, \dfrac{a}{b}, a$$ Determine the value of $b$ for which this is an arithmetic progression ($a \neq 0$) I don't know how to do this. I've tried ...
1
vote
3answers
39 views

Find the series expansion of 2 multiplied functions

The first three terms in the series expansion of $(1+x)^m$ are $1 + mx + \dfrac{m(m-1)x^2}{2}$. Find the first 3 terms in the series expansion of $(1+x)^{m+1}(1-2x)^m$. I don't really know ...
2
votes
1answer
67 views

The sequence $H_n-\ln(n)$ converges

Is there a proof that the sequence $\displaystyle \sum_{k=1}^n \frac{1}{k}-\ln(n)$ converges that doesn't use integrals?