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

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0
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1answer
4 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 ...
-4
votes
1answer
14 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?
17
votes
1answer
189 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
43 views

Paradox or error in design?

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 ...
-2
votes
1answer
23 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
28 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 ...
4
votes
2answers
34 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
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0answers
11 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
50 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
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0answers
10 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
45 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
63 views

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

First of all , I'll let you know that I am really really bad at calculus so please be gentle. Lets have this series: $\sum_{n=0}^\infty \frac{(-1)^{n-1}x^{2n}}{n(2n-1)}$ The thing is I know ...
3
votes
1answer
24 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
80 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
15 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
32 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
24 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
11 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
88 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
37 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
37 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 ...
-6
votes
3answers
102 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
109 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
16 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
44 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
41 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 ...
1
vote
1answer
21 views

uniformly convergent sequence of differentiable functions, series of derivative of terms not convergent

I am attempting to come up with a uniformly convergent sequence of differentiable functions $g_{n}:(0,1)\to \mathbb R$ such that the sequence $\{g_{n}'\}$ does not converge. I was thinking that ...
1
vote
1answer
69 views

Generating function for any series

Given a summation series, is there any way to generate a function to find the value of the sum of first n terms? For example, we have, $\sum f(n) = f(0) + f(1) + ... + f(n)$ . Now, I want to know ...
-1
votes
1answer
38 views

What will I pay in month x if I pay 1/36 of balance each month? [on hold]

I have an open note at a bank that I pay 1/36th of the balance every month. I am looking for an equation that will allow me to know what my payment will be on x month.
0
votes
0answers
31 views

Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x)

Here is the theorem in mathjax: A real value function $f$ is continuous at $x \in R$ iff whenever a sequence of real numbers $x_{n}$ converges to $x$ then the sequence $f(x_{n})$ $\rightarrow f(x)$. ...
4
votes
0answers
72 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
6
votes
8answers
4k views

1, 5, 9, 13, 17, 21,…

How would you describe the set $\{1, 5, 9, 13, 17, 21,\dotsc\}$ in the style of $x:P(x)=$? I know that the sequence is "the last number + 4" or $4n-3$.
9
votes
3answers
199 views

If $u_{n+1}\le u_n+u_n^2$ and $\sum u_n$ converges, prove that $\lim\limits_{n\to +\infty}(n\cdot u_n)=0$

Given the positive sequence $\{u_n\},n\in \mathbb{N}$ that meets the conditions: $\boxed{1}$. $u_{n+1}\le u_n+u_n^2$ $\boxed{2}$. Exist the constant $\text{M} >0$ so that ...
1
vote
3answers
103 views

Find the series: $\frac{-1}{4}+\left(\frac12+\frac14+\frac28+\frac3{16}+\frac5{32}+\cdots\right)$

Find the series: $$\frac{-1}{4}+\left(\frac12+\frac14+\frac28+\frac3{16}+\frac5{32}+\cdots\right)$$ Evidently, this is a Fibonacci Sequence with a Geometric Sequence. But I don't think there is a ...
0
votes
0answers
16 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
2
votes
1answer
35 views

How to calculate the closed form of the Euler Sums

We know that the closed form of the series $$\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}\left( {\begin{array}{*{20}{c}} {2n} \\ n \\ \end{array}} \right)}}} = \frac{1}{3}\zeta \left( 2 ...
-1
votes
2answers
73 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
1
vote
1answer
12 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
0
votes
1answer
34 views

Series summation of Geometric-Harmonic series

I am trying to find the series summation for the following series : $ \sum_{i=0}^{k} \frac{\beta^i}{i+1}$ and $ \sum_{i=0}^{k} \frac{\beta^i}{(i+1)^2}$ $\beta \in (0,1)$ Any ideas on how to ...
-1
votes
1answer
88 views

What is the next number of the following sequence 27, 54, 81, 135, 189,…

What is the next number of the following sequence 27, 54, 81, 135, 189,........ Options Given: 1) 108 2) 243 3) 405 4) 216 5) 378 6) 486 7) 297 8) 459 9) 351 10)None of these My Approach: ...
1
vote
1answer
28 views

What is the Limit of the following Fibonacci Sequence?

The Fibonacci numbers $x_1,x_2,.......,$ are defined recursively by $x_1=1, x_2=2$ and $x_{n+1}=x_n+x_{n-1}$ for $n\geq2$. Show that, $\lim_{n \to \infty}\frac{x_{n+1}}{x_n}$ exists, and evaluate the ...
2
votes
0answers
19 views

The remainder estimate for the integral test

The remainder estimate for the integral test states that if $a_k=f(k)$ where $f$ is a continuous, positive, and decreasing function on $[n,\infty)$ and $R_n=s-s_n$ (where $s_n$ is the $n$th partial ...
0
votes
1answer
68 views

what's the limit $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2}$

I have no idea how to compute the tail sum $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2} $. I tried subtracting the first n items from all but realized that I don't know a way to calculate ...
3
votes
4answers
89 views

Proving that all terms in sequence are positive integers (Recursive)

I am preparing for Putnam and working through practice-like questions and I am boggled. I've got the sequence defined by $a_1 = 1$ and $a_{n+1}$ = $2a_n+\sqrt{3a_n^2-2}$ for any $n\in\mathbb{N}$ and ...
0
votes
1answer
21 views

Proving convergence of a sequence through a polynomial

Let ($x_n$)$\rightarrow$x and let $p(x)$ be a polynomial. (a) Show $p(x_n)$$\rightarrow$$p(x)$. (b) Find an example of a function $f(x)$ and a convergent sequence ($x_n$)$\rightarrow$x where the ...
1
vote
1answer
81 views

Analytic representation of Harmonic numbers

As we know, using $$\frac{{{{\ln }^2}\left( {1 - x} \right)}}{{1 - x}} = \sum\limits_{n = 1}^\infty {\left( {H_n^2 - {\zeta _n}\left( 2 \right)} \right){x^n}} = \sum\limits_{n = 1}^\infty {\left( ...
1
vote
1answer
23 views

Convergence Proof Help?

Question: Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence $(x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ ...
3
votes
1answer
62 views

Can the Fibonacci sequence be written as an explicit rule?

When I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = ...