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

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5
votes
1answer
22 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 $\operatorname{var}(T)$. My work: I find $T$ is a UMVUE ...
-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 ...
10
votes
4answers
243 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
votes
0answers
26 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.
1
vote
3answers
72 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
votes
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 ...
8
votes
2answers
234 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
-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
1answer
46 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 ...
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
583 views

compound interest with geometric series

Were studying geometric sequences in maths and this came up as one of the questions: A mortgage is taken out for 150000 and is repaid annually with 20000 installments. Interest is charged on the ...
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
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 ...
4
votes
1answer
88 views

What rotation rules can be applied to stacked cubes to make a 3D spirograph?

If you arrange building blocks for example toy cubes so that every next cube is tilted over its base by 30 degrees and rotated to it's right by 12 degrees, it would wind through space in a helical ...
-4
votes
2answers
222 views

Finding radius of convergence for series

Find the radius of convergence of the given power series: $$\sum _{n=1}^{ \infty} \frac{(-1)^n x^{5n+2}}{5n+2}$$ Through the ratio test, I can get it down to $$ \frac{(-1)^n+1 ...
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 $146$. ...
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 ...
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
votes
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 ...
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 ...
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 ...
1
vote
1answer
46 views

solving a linear recurrence relation simple moving average

Here's a recurrence relation, $k$ is fixed: $$\frac{1}{k}\sum_{n=i}^{k+i-1} a_n = a_{k+i}$$ for all $i\in \mathbb{N}$, and for $a_i$ with $1\leq i \leq k$ we have fixed non-negative real number ...
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 ...
4
votes
4answers
132 views

Which (convergent) series can one find the sum of?

I know about geometric series and how one can find the sum when they are convergent. I also have heard that one can prove that the $p$-series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ has sum $\pi^2/6$ ...
-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 = ...
6
votes
1answer
294 views

Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?

If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has $$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
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 ...
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
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
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 $\{f_n\}$ ...
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 $\pi$. ...
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?
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 ...
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
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 ...
11
votes
3answers
179 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
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 ...
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 ...
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 $k<0$ ...
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 ...
-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 ...
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: ...
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?
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 ...