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

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6
votes
3answers
107 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
3answers
63 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
95 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
81 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
46 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
19 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
35 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
19 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
64 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
109 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
50 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
39 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
34 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
48 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
215 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
94 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 $$\left(\frac{7}{4}\right)^{2n}$$ ...
6
votes
3answers
511 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
37 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
58 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: $$\lim_{n\to+\infty}\frac{a_n}{\log n}=1.$$
4
votes
1answer
36 views

What is this sequence of all permutations with gaps permissible

Let there be a sequence $a_1, a_2, a_3,...,a_n$ that represent some actions that you know are required to solve a problem. However, you do not know what order these actions need to be taken to solve ...
1
vote
3answers
48 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
5
votes
3answers
584 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
1
vote
2answers
130 views

Why $\sum_{n=1}^\infty \frac{1}{2^{n}\ n} =\log (2)$?

Could someone explain me this equation? $$\sum_{n=1}^\infty \frac{1}{2^{n}\ n} =\log (2)$$ I am to calculate the sum of the expression on the left side. I was looking for mathematical formulas, any ...
2
votes
2answers
52 views

Convergence/Divergence of a the series $\sum_{k=1}^{\infty} a_k$, where $a_1=1$ and $\forall 1\leq k\in\mathbb{N},a_{k+1}=\cos(a_k)$

I got this question: Determine wether the series $\sum_{k=1}^{\infty} a_k$ absolutely converges, conditionally converges or diverges, where $a_1=1$ and for each $1\leq k \in\mathbb{N}$, ...
0
votes
1answer
39 views

Best approach or algorithm to solve equation with multiple variables?

I have an equation : $A^6x_1 + A^5x_2 + A^4x_3 + A^3x_4 + A^2x_5 + A^1x_6 + x_7 = B$ What can be the best algorithm/approach I can use to crack this? $A$ and $B$ are constants. $x_1,x_2...x_7$ are ...
2
votes
3answers
75 views

Evaluating the alternating series $\sum_{n = 1}^\infty \frac {(-1)^{n-1}}{3^{n-1}}$

How to find such alternating series sum? \begin{equation} \sum_{n = 1}^\infty \frac {(-1)^{n-1}}{3^{n-1}} \end{equation}
0
votes
2answers
37 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
1
vote
4answers
73 views

Show that $\langle f_n \rangle$, where $f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$ is a Cauchy sequence.

Show that $\langle f_n \rangle$, where $$f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence. My attempt: Consider $$|f_{2m}-f_m| = \left| ...
2
votes
4answers
67 views

Convergence of $a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$

Show that the sequence $$a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$$ does not converge but the sequence $b_n=\frac{a_n}{n}$ converges. I can show the first part. For the second part, will it be sufficient ...
0
votes
0answers
38 views

How find $x\in\mathbb{R}$ such $f_n(x)=\sin(7^n\pi x)$ is converges? [on hold]

How find all $x\in\mathbb{R}$ such $f_n(x)=\sin(7^n\pi x)$ is converges? Find this limit.
7
votes
4answers
144 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
0
votes
0answers
30 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
3
votes
1answer
86 views

Evaluating an infinite square root

How do I evaluate the square root: $$\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\cdots}}}$$ I have tried creating two arithmetic sequences such that $$a_n = 1999+14n$$ $$b_n = 274+2n$$ so the square ...
0
votes
4answers
156 views

Find whether the following series converges or diverges $\sum_{n=1}^{\infty}\frac{\ln n }{\sqrt{n}}$ [on hold]

Looking for a witty answer. I can see that the given series converges by AST. All Ideas Appreciated
0
votes
0answers
13 views

proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
4
votes
0answers
36 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
1
vote
2answers
49 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
1answer
19 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
16
votes
1answer
160 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
3
votes
2answers
52 views

Proof $e^n*n!$ is an asymptote of $(n+1)^n$

I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$. All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
0
votes
1answer
27 views

Calculate the supremum of $\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$

If $\{\gamma_n\}$ is a sequence of real number and $\exists M>0$, finite, such that $|\gamma_n|\leq M$, find the supremum of the following sequence: $$\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$$
4
votes
0answers
50 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
1
vote
1answer
27 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
0
votes
1answer
31 views

Simpler way of proving series convergence?

Determine whether the following series converges $$\sum_{n=1}^{\infty} \left (\frac{n^4}{n^4 + 2}\right)^{n^5-3}.$$ I've found convergence using the root criterion in the following way. $\sqrt[n]{ ...
1
vote
2answers
72 views

$\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges.

Assume that $\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges. Same to be easy at first thought, can ...