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

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0answers
24 views

Convergence of $\frac{1}{4n^2-1}$ on the range of n=1 to infinity.

I know that the series converges to $\frac{1}{2}$. Can someone please show the steps to prove this though and what convergence test can be used? Thanks.
5
votes
1answer
21 views

Find $\sum_{k=0}^{\infty}(1-1/n)^{2k}\frac{e^{-n\theta}(n\theta)^{k}}{k!}$ (the variance of $(1-1/n)^{X_1+\cdots+X_n}$)

Given a random sample $X_1,\ldots,X_n$ from Poisson distribution with an unknown parameter $\theta>0$.$T:=(1-1/n)^{X_1+\cdots+X_n}$. Find $var(T)$. My work: I find $T$ is a UMVUE of ...
1
vote
3answers
71 views

Does every irrational number contain arbitrarily long sequences of some digit?

Suppose we have an irrational number represented in base 3 such that all sequences of 1's and 2's in its ternary expansion have length less than some $n $. Does this imply there are arbitrarily long ...
0
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0answers
23 views

Question on Sum of a recursive sequence. [duplicate]

I'm re-posting as my previous post didn't serve me with any answers. Atleast tell me the approach. Tnx in advance. :) $$\sum_{r=1}^n U_r$$ where $$U_r = \frac{U_{r-1}M_r}{M_{r-1}(a+b M_r)}$$ and ...
-4
votes
0answers
59 views

Help me sum up this series

I stuck up at this stage while trying to prove a physical chemistry equation. please help me. $$\sum_{r=1}^n U_r$$ where $$U_r = \frac{U_{r-1}M_r}{M_{r-1}(a+b M_r)}$$ and $$U_1 = \frac{M_1}{a+b M_1} ...
3
votes
4answers
69 views

The sequence $x_{n+1}=ax_{n}+b $ converges to where?

$$a,b \in \mathbb R , \ 0\lt a\lt 1 . $$ Define the sequence $$x_{n+1}=ax_{n}+b \text{ for } n\ge0\ .$$ Then for a given $\ \ x_0\ \ $ , does this sequence converge? And if it does, to ...
0
votes
1answer
24 views

Fibonacci sequence in system of equations?

Can we write/solve the fibonacci sequence in a linear system of equations, for a given number of terms? I know we can define the recursive definition using matrices but what i am interested in is ...
-1
votes
0answers
25 views

$n$th derivative of $\sin(nx)$? [on hold]

Don't know how to solve this one please answer in a detailed manner. Thanks in advance... Answer is given in a recursive form. I have thought of using the expansion of $\sin(nx)$ and then ...
-1
votes
1answer
64 views

Find the $nth$ derivative of $y=\sin(x^2)$ [on hold]

Please help me and its not a homework.I have tried it a hell lot of many times and even asked my seniors but none had solved it ...So plzzz help me know step by step how to solve it..thanks in advance ...
0
votes
1answer
11 views

Limit of sequence partially applied to a function

Let $(a_n)_{n\in\mathbb N}$, $(b_n)_{n\in\mathbb N}$, $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ be real-valued sequences and $f:\mathbb R\to \mathbb R$ monotone with ...
1
vote
2answers
29 views

At least how many numbers should be selected from the set {1, 5, 9, 13, …125} to be assured that two of the numbers selected have a sum of 146?

I know the answer is 20 (says the answer key), but I'm not quite sure how it got it. I also know that in the sequence, we can pair $21+125$, $25+121$, $29+117$... and so on to get a sum of $146$. ...
0
votes
3answers
48 views

¿What mistakes can be made when differentiating power series (e.g $\sin x$ power series)?

I know that the derivative of $\sin x$ is $\cos x$, but I don't know what is wrong with the following: $$\sin x = \sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}.$$ Now if I want to find its ...
0
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0answers
18 views

Absolute convergence related to Fourier analysis

If the Fourier transform of a function $f \in L^2$, whose frequency $\xi$ satisfies $|\xi| \leq \pi$, has compact support, it is famous that \begin{align*} f(x) = \sum_n ...
0
votes
1answer
48 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
1answer
63 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
47 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
96 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
16 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
42 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
32 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
9 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
59 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
19 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
51 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
378 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
66 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
31 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
55 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
56 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
38 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: ...
6
votes
3answers
125 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 ...
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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?
23
votes
2answers
662 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 ...
4
votes
2answers
125 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 ...
-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
30 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
41 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
39 views
+50

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
49 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?
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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 $$ H_n \leq H_{\left\lfloor\frac{n}{2}\right\rfloor}+1.$$