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

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0
votes
1answer
41 views

Proving $x_ky_k\to ab $

Prove that the sequence $x_ky_k\to ab $ if $x_k\to a$ and $y_k\to b$. I wanted to try and do this with the epsilon definition but i am having a few technical issues. Proof: $$ |x_ky_k - ab| = |x_k ...
3
votes
2answers
50 views

Real Analysis, Cauchy but not null.

I came across this question in a book on p-adic numbers and thought it looked interesting. However, I am having trouble getting started with it. Any hints/suggestions is much welcomed Let $(a_n)$ be ...
0
votes
2answers
42 views

Help with this mathematical induction please? [on hold]

Use mathematical induction to prove that the following is true for every positive integer $n$: $$\dfrac{1}{4}+\left ...
0
votes
1answer
89 views

How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test?

How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test? I tried using the comparison test but I could not come up with an inequality that helps ...
-1
votes
0answers
14 views

Finding the sum of a series derived by Nernst equation

I was working on Nernst equation for non-electrolytic solutions and came across this series. Please help me find the sum to the nth term. I present you the most general term I could. $$U_r = ...
2
votes
1answer
12 views

Investigating uniform convergence of a sequence

I am trying to determine if the sequence $f_n$:= $\frac{x^{2n}}{1+x^{2n}}$ is uniformly convergent on $D_1:=[-q,q],0<q<1$, and $D_2:= (-\infty,r] \cup [r,\infty),r>1$. I have determined that ...
1
vote
1answer
40 views

Is there a geometric progression containing 2, 3 and 5

I am trying to find a geometric progression containing 2, 3 and 5 (the terms do not have to be consecutive). If there is no such progression, is it possible to prove this? Thanks in advance.
1
vote
2answers
29 views

Sequence of functions and function series

For every $n \ge 0$ we define function $f_n:[-1;1]\rightarrow \mathbb{R}$ $f_n(x) = \sqrt[n+1]{n+1}(\frac{x+x^2}{2})^n$, $x\in[-1;1]$, $0^0=1$. a)determine whether sequence of functions $\{f_n\}$ ...
0
votes
0answers
8 views

Representing trigonometric functions in a form of rational functions

Here I introduced a non-Archimedean numerical system in which the real numbers are extended by elements $\omega_-$, $\tau=\omega_-+1/2$, $\omega_+=\omega_-+1$ in such a way that standard parts of ...
6
votes
1answer
56 views

how do I find the general term here?

I am getting crazy on this series! I found this in a handwritten old book without a reference. I could not figure out how it is built but the series numerically seems to converge to $\pi$. ...
1
vote
1answer
18 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
1
vote
2answers
35 views

Finite series identity [duplicate]

How would I prove this statement? I know that it's a finite series. I don't know how to approach this at all. $$\sum_{i=1}^N i^3 = \left(\sum_{i=1}^N i \right)^2$$
3
votes
3answers
49 views

Sum of infinite geometric series

How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus. $$\sum_{i=0}^\infty \frac{i}{4^i}$$
0
votes
0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
6
votes
4answers
370 views

How to prove $\sum _{k=1}^{\infty} \frac{k-1}{2 k (1+k) (1+2 k)}=\log_e 8-2$?

How to prove: $$\sum _{k=1}^{\infty} \frac{k-1}{2 k (1+k) (1+2 k)}=\log_e 8-2$$ Is it possible to convert it into a finite integral?
2
votes
1answer
56 views

positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove$\displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent

When I do my homework ,I met this problem: If positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove:$ \displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent. I ...
1
vote
1answer
30 views

Divergent or convergent but how ??

I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...
3
votes
4answers
51 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then ...
-3
votes
2answers
55 views

How to calculate $\frac1n$ sequence? [on hold]

How to calculate the following sequence? $$E(n) = \frac 1{1 \cdot 4} + \frac 1{4 \cdot 7} + \ldots + \frac 1{(3n-2)(3n+1)}; n \in \mathbb N $$ a) calculate for $$E(2006)$$ b) proove that $$E(n) \in ...
0
votes
1answer
35 views

Find the missing number in the series?

In the given series , find the missing number in the given series :13,14,22,49,113,___,454?
2
votes
0answers
23 views

What do the Stirling numbers of the first kind have to do with polylogarithms?

On a whim, I had decided to look into ways of evaluating series of the form $$\sum_{n\ge1}\frac{1}{n^k2^n}$$ which I learned has a more general form in terms of polylogarithms: ...
5
votes
3answers
112 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
0
votes
1answer
8 views

Showing a proposition of sequence

How would I show the following If limit $j\rightarrow \infty$ for the sequence $b_j=B$ and B<0 then there exist an number N in natural number such that when j>N then $b_j<0$ Would I start ...
-3
votes
0answers
19 views

Find $ a_{n}$ and prove these are geometric sequences [on hold]

Given $$ S_{n} = 2^{n+3} - 8 $$ How do I find $a_{n}$ and prove that the sequence is geometric?
22
votes
2answers
644 views

An integral identity from Ramanujan's notebooks

Browsing through Ramanujan's notebooks, I found the following identity, without proof of course (Notebook 1, p. 130): In other words (took me a while to realize that the lower integration bound is ...
0
votes
2answers
49 views

Paradox or error in design? [on hold]

Currently I'm writing a homework for my school. I've made an experiment built this way: There is a laser pointed at a half reflecting mirror which reflects 50% at a wall. The other half cross the ...
-1
votes
1answer
26 views

Regarding Power series in complex analysis [on hold]

Suppose that I have a series $\sum_n^{\infty} \frac{z^n}{n}$.It is convergent for $|z|<1$. I want to know why the above series converges for $|z|=1$ except at $z=1$.
0
votes
1answer
29 views

infinite series sum can't find the geometric series: $\sum_{i=0}^\infty (2^i +4^i)/6^i $

$$\sum_{i=0}^\infty \frac{2^i +4^i}{6^i} $$ I'm not able to get a geometric series out of this. If I can the geometric series ,the infinite summation from there is easy
1
vote
0answers
15 views

Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums?

I asked in a previous question whether a function, $a_n$, is unique to $F(s)$ for any Dirichlet function defined by the following $$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$ Its uniqueness property ...
5
votes
2answers
39 views

Sequence of equations

The sequence continues infinitely, why do the equations below work? $$1+2=3$$ $$4+5+6=7+8$$ $$9+10+11+12=13+14+15$$ So I've been trying to observe some patterns but none seem to help me. So I ...
2
votes
0answers
22 views

A functional equation relating two harmonic sums.

Introduction. I computed two Mellin transforms while browsing / working on the problem at this MSE link. No solution was found, but some interesting auxiliary results appeared. I am writing to ask for ...
3
votes
1answer
55 views

Is there a nice expression for $f(x) = (1+x)(1+x^2)(1+x^3)\cdots$

While I was solving a problem, I stumbled upon this function $$f(x) = (1+x)(1+x^2)(1+x^3)\cdots$$ I tried to write out the first few products but I couldn't recognize any meaningful pattern. Is there ...
0
votes
0answers
11 views

Kloosterman Sums and Lattice Hyperbolas

Part of this blog discussing the twin prime conjecture mentions a connection between three objects: $ \sum_{x \leq n \leq 2x} \tau\Big(n(n+2)\Big) $ average over twin primes where $\tau(n) = (1 ...
0
votes
1answer
48 views

Finding patterns in seemingly arbitrary pairs of numbers

I don't work (directly) in mathematics (I'm a programmer), but I see numbers every day. Today I came across an issue where some totals were off, and was sent a list of the last 9 examples of the ...
0
votes
2answers
76 views

Why is convergence required for a series to be differentiable? [on hold]

Since moderators marked this question as "unclear" I will repeat the title maybe this won't be marked. Why is convergence required for series to be differentiable? I want intuitive explanation - not ...
3
votes
1answer
26 views

Discrete analogue of bounded variation

What kind of sequences $(a_n)\subset\mathbb{R}$ are expressible as the difference of two increasing sequences?
-4
votes
1answer
84 views

Show that $ H_n \leq H_{\left\lfloor\frac{n}{2}\right\rfloor}+1$ [on hold]

Consider the harmonic sequence $$H_n = 1 + \frac{1}{2} + \frac{1}{3} +\frac{1}{4} + \ldots + \frac{1}{n}$$ I would like to prove that $$ H_n \leq H_{\left\lfloor\frac{n}{2}\right\rfloor}+1.$$
0
votes
0answers
16 views

Average of Convergent Sequences Proof [duplicate]

Show that if ($x_n$) is a convergent sequence, then the sequence given by the averages: $y_n$=($x_1$+$x_2$+...+$x_n$)/n also converges to the same limit. I know that for all $\epsilon$$>$0, ...
2
votes
1answer
35 views

Floor and Ceiling Series (I) [on hold]

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
2
votes
0answers
25 views

Infinite Bessel function sum

Let $$f(x)=\sum_{p=0}^\infty J_p(x)$$ We have that $f(0)=1$ and a Bessel function like curve. Can $f(x)$ be expressed in terms of known functions? $$f(x)=1-\sum_{n=0}^\infty ...
6
votes
0answers
28 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
0
votes
0answers
12 views

Help in estimating error in alternating series. (homework)

I tried to do it (4 times already actually) I read that to get the error (upper bound) I should get the value of a(n+1) which in this problem is the value of the term at n=23. But I do not know why am ...
6
votes
3answers
91 views

If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$

I found an interesting conclusion when I did this simple question. Let $$f(x)=(x^2-1)(x+2)=x^3+2x^2-x-2$$ and let $x_i$ for $i=1,2,3$ be the roots of $f(x)$. Find the sum $\sum\limits_{i=1}^3x_i^2$. ...
3
votes
1answer
41 views

Planar Graph Isomorphism

In 1980, I. S. Filotti & Jack N. Mayer proved planar graph isomorphism testing could be done in polynomial time. Does anyone have an implementation of that? I have a few billion planar graphs ...
1
vote
0answers
38 views

The sum of the infinite series [duplicate]

The sum of the infinite series: $$ \frac{1}{2} +\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +\cdots$$ I am able to find ...
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votes
3answers
109 views

What is solution to this maths series problem? [on hold]

I found this question on facebook and me and my friend were discussing the possible solution for 9. We have found 3 answers and none of us has any idea which one is correct as all of them looks ...
4
votes
4answers
113 views

Find $\sum\limits_{n=1}^{\infty}\frac{n^4}{4^n}$

So, yes, I could not do anything except observing that in the denominators, there is a geometric progression and in the numerator, $1^4+2^4+3^4+\cdots$. Edit: I don't want the proof of it for ...
0
votes
1answer
17 views

Find a geometric progression with sum $100$ [on hold]

I have to find an infinite geometric progression having sum $100$, then to find its first term by assuming that the common ratio is $\frac{1}{4}$. Any hints?
2
votes
0answers
57 views

prove this sequence to decreasing for all $n$

Define $a_{n}=1$,and such $$a_{n+1}=a_{n}+\ln{\left(\dfrac{n}{n+1}\right)}+\dfrac{1}{e^{a_{n}}\cdot n}$$ show that $$a_{n+1}<a_{n}$$ or ...
0
votes
1answer
43 views

Vector Space that is a Sequence Space

Consider the sequence $(x_n)_{n \in \Bbb N}$ $ |$ $ x_n=\sum_{i=0}^{n}f(i)$ then if $f(i)$ can also be viewed as a sequence $(y_n)_{n \in \Bbb N}$ $ |$ $y_n=f(n)$. Both of these sequences ...