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

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2
votes
2answers
37 views

Finding the convergence

The sequence $a_n = \sqrt[n]{(3^n+5^n)}$ is convergent? Tried to resolve applying the the limit $n\to\infty$ but couldn't figure out how to finish it, any idea? Thanks
1
vote
4answers
165 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
0
votes
0answers
17 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
0
votes
0answers
14 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta$, where a and b are real numbers. If we consider ...
0
votes
1answer
21 views

$\ell^p$ spaces' inclusion

$$ \ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q $$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
0
votes
0answers
47 views

Does the sequence is convergent? [on hold]

Prove or disprove the convergence of sequence $(v_n )_{n=1}^{\infty}$ where $$v_n =n\sum_{k=1}^{\infty} \frac{1}{2^k }\left(1-\frac{1}{2^k}\right)^n .$$
1
vote
4answers
57 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
0
votes
4answers
56 views

How to prove {$a_n$ } is increasing where $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{ 2+ a_n}$ [duplicate]

I already found out that this sequence is bounded above and $a_n <2 \forall n \in \mathbb Z_+ $ I think I'm missing a point as I can't think of a way to prove that the sequence is increasing.
0
votes
2answers
47 views

If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too?

I got this question: Prove or disprove the following: If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too? I tried to find a couple of ...
1
vote
1answer
26 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
1
vote
2answers
43 views

Proving that a convergent sequence has a unique limit

Is the following method wrong? Let {$a_n$} be a convergent sequence Assume $ \lim_{n \rightarrow \infty} \{a_n \}$ = L and $ \lim_{n \rightarrow \infty} \{a_n \}$ = M L-M =$ \lim_{n \rightarrow ...
3
votes
0answers
38 views

Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
2
votes
4answers
48 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
0
votes
0answers
13 views

Recurrence relation-Summation of a series [on hold]

Sir, I have a Converging recurrence relation given as below, $-(\psi(n-1)) ...
6
votes
2answers
78 views

How to simplify this summation

I was wondering how to solve this infinite sum. $$\sum_{k=0}^\infty {1\over 4!} \cdot {k^7\over2^k}$$ I know roughly that for $$\sum_{k=0}^\infty {k \over 2^k}$$ the sum takes advantage of the ...
0
votes
1answer
36 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
0
votes
0answers
23 views

convergence interval of an infinite series without the general term

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...
2
votes
1answer
89 views

Simplify this summation

$$\sum_{k=5}^\infty{{k-1}\choose{k-5}}\frac{k^3}{2^k}$$ I can't seem to simplify this sum. I get to a certain point then I get stuck, I know there must be some sort of trick to simplify it but I am ...
0
votes
0answers
61 views

Sum of series with two binomial coefficients [on hold]

How can I find sum of a series $$\sum_{k=m}^{\infty} {k \choose m} {k+n \choose n}x^k$$ where $x<1$ and $m,n$ are constants. Using wolfram mathematica it is obtained $$\sum_{k=m}^{\infty} {k ...
0
votes
0answers
27 views

Equality of a recurrent sequence and of a running maximum of another sequence

Let $\{a_n\}$ be a sequence of real numbers. Let $c,b$ be real constants. Define $$ L_{k,n}=\exp\left\{c\sum_{i=k}^n(a_i+b)\right\}. $$ Then it can be shown that $L_n=\max_{1\le k\le n}L_{k,n}$ is ...
2
votes
6answers
198 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
1
vote
2answers
57 views

Prove or disprove $\sum (a_n + b_n) $ is divergent if $\sum a_n $ and $\sum b_n $ are divergent.

I proved it as follows. Since $\sum a_n$ and $ \sum b_n $ are divergent, $ \forall \epsilon > 0, \exists p \in \mathbb Z_+ st, n \gt p \implies \sum a_n > \epsilon \gt \frac{\epsilon}{2} $ ...
0
votes
0answers
54 views

Logic of numerical series

One of our colleagues has written a numerical series to the whiteboard in our breakroom. Nobody until now could solve the logic behind this series: ...
1
vote
0answers
44 views

Convergence/divergence of $\sum_{n=2}^{ \infty} [ (1+\frac{1}{\log n } )^{1/n}-1 ]$ [on hold]

Determine whether $$\sum_{n=2}^{ \infty} \left[ \left(1+\frac{1}{\log n } \right)^{\large\frac1n}-1 \right]$$is divergent or not. I tried the ratio test and the limit comparison, but they did not ...
1
vote
1answer
40 views

How to define divergence and prove $\sum r.a_n$ is divergent if $\sum a_n$ is divergent [on hold]

The first question was to prove or disprove $\sum ra_n$ is divergent if $\sum a_n $ is divergent (r $\in \mathbb R$ and r$\neq$0). I came across another problem when I was trying it, can I write the ...
6
votes
3answers
197 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
2
votes
4answers
97 views

Why these two series are convergent or divergent?

I do not understand why $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac1k$$ is divergent but the other series $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac{(-1)^{k+1}}k$$ is convergent. ...
12
votes
1answer
103 views

A series involving inverses of harmonic numbers

How would I solve this question: If $$E_n = \frac{1}{2} + \frac{1}{4} + \frac{1}{6}+ \cdots +\frac{1}{2n}$$ and $$A_n = (2n+1)(E_n)(E_{n+1})$$ Find $$\sum_{n = 1}^{\infty}\frac{1}{A_n}$$ My try: ...
2
votes
2answers
106 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
3
votes
0answers
64 views

How do you use the BBP Formula to calculate the nth digit of π?

I know what the Bailey-Borweim-Plouffe Formula (BBP Formula) is—it's $\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} ...
1
vote
1answer
21 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
1
vote
2answers
24 views

Finding common ratio from two sums

I'm struggling with this very basic question on the binomial theorem: The sum of the first and second terms of a geometric progression is 12, and the sum of the third and fourth term is 48. Find ...
1
vote
1answer
38 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
0
votes
1answer
20 views

Series of infinite terms where individual terms are multiplied by the order of the term

I would like to know what the equation is for as series of infinite terms which are multiplied by the order of the terms: $$ \sum_{i=0}^{\infty} \sum_{j=0}^{\infty}(ij) a^ib^j $$ $a$ and $b$ are both ...
1
vote
3answers
43 views

A question on arithmetic progression [on hold]

If $a\left(\frac{1}{b} + \frac{1}{c}\right), b\left(\frac{1}{a} + \frac{1}{c}\right),$ and $c\left(\frac{1}{a} + \frac{1}{b}\right)$ are in arithmetic progression, then prove that $a,b,$ and $c$ are ...
4
votes
2answers
66 views

How to prove that $ 1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$

How would I prove this inequality (assuming its true, its from a textbook) $$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$ if $n > |x|$, $x\in R$ and $n\in N$ I first ...
0
votes
1answer
113 views

Calculating of the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$? [on hold]

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
-1
votes
2answers
55 views

Maclaurin series of the function $\frac{x^2}{2+3x^2}$

I got this question: Find the Maclaurin series of the function $\frac{x^2}{2+3x^2}$ and find its domain of convergence. I tried using the binomial series $(1+x)^m = 1 + \sum_{k=1}^{\infty}{m \choose ...
-1
votes
2answers
40 views

Product of Infinite Series

I am trying to compute the product of 3 infinite series. As such, I need the compact form for the product ...
1
vote
0answers
35 views

A sequence defined as $a(n)=n-a(a(n-1))$ $n\geq 1,\ a(0)=0$, how to prove that $a(n)=⌊(n+1)(-1+√5)/2⌋$

$a(n)=n-a(a(n-1)), \ n \geq 1,\ a(0)=0$, to prove that $$ a(n)=⌊(n+1)\cdot \frac{\sqrt{5} - 1}{2}⌋. $$ This is an exercise of "Discrete Mathematics and Its Application".(Supplementary exercise 72 of ...
2
votes
2answers
49 views

The Result of Dividing 2 Power Series

Is there a way to write a single series for the following division? $$\frac{\displaystyle\sum_{i=0}^{\infty}a_{i}x^{i}}{\displaystyle\sum_{j=0}^{\infty}(j-2)a_{j}x^{j}}$$ Thanks, Radz.
2
votes
3answers
220 views

Summation of Infinite Geometric Series

Determine the sum of the following series: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$ My work: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} ...
2
votes
2answers
95 views

How to I write $\frac{7^{2n}}{4^{3n}}$ as a geometric series?

I am trying to write $$\frac{7^{2n}}{4^{3n}}$$ as a geometric series which has the form:$$\sum\limits_{i=0}^n{ar^n}$$. I'm not sure if I should get in the form $$\left(\frac{7}{4}\right)^{2n}$$ ...
0
votes
1answer
60 views

Every Cauchy sequence is bounded

Please help me to understand step-by-step how this example is proven. The statement is follows: Every Cauchy sequence is bounded. since I do not understand how verified, please help me, thank ...
6
votes
3answers
535 views

How do you calculate this sum?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
3
votes
4answers
59 views

Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
0
votes
1answer
38 views

Generalized Fibonacci Sequence

I'm having trouble with a problem I encountered while studying Number Theory. This problem comes from the book Number Theory by George E. Andrews. It defines a generalized Fibonacci sequence $F_1$, ...
1
vote
1answer
64 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
8
votes
1answer
100 views

Feynman's Algorithm for computing a logarithm of a number in [1,2]

I came upon the following quote concerning Feynman (the entire essay this is from can be found here): Consider the problem of finding the logarithm of a fractional number between 1.0 and 2.0 (the ...
1
vote
2answers
52 views

Limit of a recursively defined sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that: $$\forall n\in\mathbb{N},\quad a_{n+1}=a_n + e^{-a_n}.$$ Prove that: $$\lim_{n\to+\infty}\frac{a_n}{\log n}=1.$$