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

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3
votes
1answer
46 views

$\sum_{n\geq 0}a_n$ converges iff $\displaystyle \sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ converges

The last problem I posted had a wrong statement. I recovered the correct one. Let $(a_n)$ be a sequence of positive real numbers. Prove that $\sum_{n\geq 0}a_n$ converges iff $\displaystyle ...
4
votes
2answers
63 views

How to evaluate the following sum? $\sum_{i = 1}^n \left\lfloor \frac{3n-i}{2}\right\rfloor.$

What is the value of the following sum? $$\sum_{i = 1}^n \left\lfloor \dfrac{3n-i}{2}\right\rfloor.$$ Especially how to handle the sums with floors? This sum appeared while solving this problem. My ...
0
votes
0answers
30 views

A problem on equicontinuity in the extended sense

I see in a book the following claim: $f_n (t) = \int_{0}^{t}g(f_n(s))ds$ where $g$ is continuous and $\{f_n\}$ are uniformly bounded sequence of step functions. Then $\{f_n\}$ is equicontinuous in ...
2
votes
0answers
66 views

Formula for sequence of integers

I am trying to compute the Taylor series of $f(x) = \sqrt{-\ln(x)}$. I compute the derivatives of $f(x)$ and evaluate them in the point $x=1/e$. The resulting expressions have the following ...
2
votes
1answer
60 views

Evaluate $\sum\frac{1}{n^n}$

What is $\sum\frac{1}{n^n}$?? I know it is $\int{x^x}dx$ but how? And what is $\int{x^x}dx$?? My approach to solve $\int{x^x}dx$:- $x^x=e^{x\ln{x}}$ integrate this and use maccaularine series. Is ...
0
votes
2answers
52 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
1
vote
1answer
79 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
1
vote
1answer
32 views

infinite sum of bounded functions unbounded?

Let $f_i$ be a sequence of positive continuous (can be assumed to be smooth) functions on the interval $[0,1]$. Furthermore, let us assume that $\sum_{i=0}^{\infty} f_i$ is finite for all $x$ in the ...
3
votes
2answers
72 views

What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
2
votes
1answer
143 views

How prove concave sequence inequality $\left(\sum_{k=1}^{n}a_{k}\right)^2\ge\frac{3n-c}{4}\sum_{k=1}^{n}a^2_{k}$

let concave sequence $\{a_{n}\}$,such $a)_{n}\ge 0$,and such $$\dfrac{a_{i-1}+a_{i+1}}{2}\le a_{i},i=1,2,\cdots,n-1$$ where $a_{0}=0$. show that $\exists c>0$ such that for every ...
2
votes
3answers
102 views

Constructing a sequence

Given two distinct, positive real numbers, how can I use these two numbers (and their non-zero integer linear combinations) to construct a sequence converges to zero? The sequence can only be of the ...
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
-3
votes
5answers
378 views

How to solve the sequence: $87, 89, 95, 107, ?, 157$

This question appeared in a competitive exam. The question is: Q. Find the unknown term in $87,89,95,107,?,157$ 1)127 $\ \ \ \ \ \ \ \ $ 2)122 3)139 $\ \ \ \ \ \ \ \ $ ...
0
votes
2answers
31 views

Arithmetic Progression-Question

The sum of $n$ terms of an A.P. is $2n+3n^2$ Find the $r^{th}$ term. TRIED ANSWER: Sum of $n$ terms $(Sn)=3n^2+2n$ ............$(I)$ $Sn=\frac{n(2a+(n-1)d)}{2}$ $.............(from formula)$ ...
-1
votes
3answers
101 views

Arithmetic Progression-Question from Hall and Knight's Higher Algebra

The question says- Between two numbers whose sum is $\frac{13}{6}$, an even number of arithmetic means is inserted such that the sum of these means exceeds their number by unity. How many means ...
2
votes
2answers
38 views

Converting a repeating decimal

I'm studying for a test and I have a question on the following problem: Convert the repeating decimal 2,307070707 into a fraction using geometric series. I'm not sure if this is right, but this is ...
6
votes
4answers
500 views

A sine integral

The integral \begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta \end{align} is claimed to not have a closed form expression. In this view find the series solution of the ...
2
votes
0answers
73 views

Puzzle with character order

Suppose I have 3 letters a, b, c and I want to find the minimum length of a string that uses all the double combinations of the aforementioned letters. How should I do it or how are such problems ...
4
votes
3answers
118 views

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
2
votes
2answers
82 views

Proving that $\sum_{k=0}^\infty \frac{e^{ikb}-e^{ika}}{k}=i\int_a^b\frac{e^{it}}{1-e^{it}}dt$

I deleted my previous question because it was basically totally wrong. Let $a,b\in ]0,2\pi[$ Prove that $\displaystyle \sum_{k=0}^\infty ...
2
votes
3answers
67 views

Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
4
votes
1answer
69 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
2
votes
2answers
42 views

Show that $\sum_{k=1}^{\infty}(\frac{1}{2})^{k} \left(1-\frac{(-1)^k}{2}\right)$ converges.

Show that $\sum_{k=1}^{\infty}(\frac{1}{2})^{k} \left(1-\frac{(-1)^k}{2}\right)$ converges. I tried to show this by using the ration test: I have: $$\frac{(\frac{1}{2})^{k+1} ...
0
votes
1answer
34 views

Limit of a differentiable function

Have $g: \mathbb{R} \rightarrow \mathbb{R}$, differentiable at some $x$. Let $(u_n), (v_n)$ be sequences s.t. $u_n \leq x \leq v_n$ for all $n$, $u_n \neq v_n$ and $u_n,v_n \rightarrow x$ as $n ...
0
votes
1answer
108 views

Formula for the sum of the cube roots and fifth roots of first '$n$' natural numbers

Does anybody know the formula by which these two sums can be computed? At least approximately?
5
votes
4answers
200 views

How to find the following sum? $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} $

I want to calculate the sum with complex analysis (residue) $$ 1 - \frac{1}{7} + \frac{1}{9} - \frac{1}{{15}} + \frac{1}{{17}} - ... $$ $$ 1 + \sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - ...
0
votes
0answers
16 views

Analytic result for this series

I'm having troubles in assessing whether this series in two unknowns does converge to a more tractable expression. The series is: $$\sum_{j,k}^\infty j \cdot k \cdot \phi_j \cdot \mu_{j,k} = c $$ ...
0
votes
1answer
37 views

Given $W_k =f(W_{k-1})$ find $W_n= f(W_0,n)$ for specific case

I am essentially trying to create a formula to a baseline for weight loss for me given $N$ days from the beginning. I have worked out that for any given day: $W_{k} = (\frac{349}{350})W_{k-1} + ...
11
votes
2answers
207 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
0
votes
2answers
55 views

How do I prove something given 3 terms of an arithmetic sequence?

If $\frac{1}{b-a}, \frac{1}{2b}, \frac{1}{b-c}$ are the terms of an arithmetic sequence, prove that $b^2 =ac$. I have no idea where to even start. Any help would be appreciated.
1
vote
0answers
71 views

Sum of poisson random variables

Let $N, X_1, \dots , X_n$ be independent random variables. $N \sim P(\lambda) \quad (\text{Poisson distribution})$, while $X_k \sim B(p)$ (Bernoulli) Let us consider the "random" sum $S = X_1 + ...
0
votes
1answer
41 views

Complex number summation.

$$ \sum_{n=1}^N\cos(2n-1)\theta=\dfrac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0.$ Deduce that $$ \sum_{n=1}^N (2n-1)\sin\left[\dfrac {(2n-1)\pi}N\right]=-N\operatorname{cosec}\dfrac\pi ...
1
vote
0answers
51 views

Abel's Functional Equation for $L(x) = \sum x^{n}/n^{2}$

In "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" Hardy talks of Abel's Functional Equation $$L(x) + L(y) + L(xy) + L\left(\frac{x(1 - y)}{1 - xy}\right) + L\left(\frac{y(1 - ...
2
votes
2answers
48 views

Convergence of series $\sum a_rb_r$

Assuming $\sum a^2_r$ and $\sum b^2_r$ converge, can we deduce that $\sum a_rb_r$ converges? It feels like we can, but how? Using Cauchy Criterion for convergence maybe? Can you hint me? Thanks a ...
0
votes
1answer
51 views

Problem with making explicit formula from recursive formula

I'm having trouble turning this recursive formula into explicit one $a_0 = 0,\\a_{n+1} = a_n(1 - b_n) + b_n,$ where $(b_n)$ is given sequence of real numbers.
19
votes
1answer
287 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
1
vote
3answers
153 views

How do I derive the formula for the sum of squares? [duplicate]

I was going over the problem of finding the number of squares in a chessboard, and got the idea that it might be the sum of squares from $1$ to $n$. Then I searched on the internet on how to calculate ...
14
votes
1answer
355 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
4
votes
0answers
28 views

Limit of a strange sequence $a_{n} = a_{[n/2]} + a_{[n/3]} + a_{[n/6]}$ [duplicate]

Recently I came across this problem : If $a_{0} = 1$ and $a_{n} = a_{[n/2]} + a_{[n/3]} + a_{[n/6]}$ where $[x]$ denotes the greatest integer not exceeding $x$ then show that $$\lim_{n \to ...
0
votes
1answer
73 views

Understanding Recurrence Relation

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
6
votes
1answer
102 views

Hyperreals - is there a “boundary” between convergent and divergent series?

Hypearreals are equivalence classes of sequences of real numbers. Is there a hypperreal number $ h $ such that for every convergent (real) series $ \sum a_n $ we have that the equivalence class of ...
2
votes
6answers
287 views

How do I find the sum of the infinite geometric series?

$$2/3-2/9+2/27-2/81+\cdots$$ The formula is $$\mathrm{sum}= \frac{A_g}{1-r}\,.$$ To find the ratio, I did the following: $$r=\frac29\Big/\frac23$$ Then got: $$\frac29 \cdot \frac32= \frac13=r$$ and ...
3
votes
2answers
87 views

Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
1
vote
1answer
33 views

Proving convergence/divergence for $p$-series

I have an exam in Calc 2 coming up. As such, I am doing previous exams given by our current professor. However, the exams lack a solution set, so I will post the question, and the answer I wrote down ...
2
votes
1answer
72 views

Show that if $\sum_{n=1}^{\infty}a_n$ converges abs. and $(b_n)_{n \in \mathbb{N}}$ is bounded , then $\sum_{n=1}^{\infty} a_nb_n$ converges abs. [duplicate]

My attempt: Since $(b_n)_{n \in \mathbb{N}}$ is bounded it follows that there exists a bounded sequence $$(c_n)_{n \in \mathbb{N}}: 0 \leq |b_n| \leq |c_n|, \forall n \in \mathbb{N}$$ Therefore it ...
1
vote
3answers
46 views

How to prove that $ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
1
vote
0answers
64 views

Does $ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right )\right) $ converge? [duplicate]

I am trying to determine whether the $$ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right ) \right) $$ converges or not. I have tried the popular tests, but all ...
2
votes
1answer
38 views

Uniform convergence of a series on $[-1,1]$.

I am stuck with this problem. I am trying to check the uniform convergence of the series $$\sum_{n=1}^{\infty} \dfrac{-2x(1+x)^{n-1}}{[1+(1+x)^{n-1}][1+(1+x)^n]}$$ on the interval $[-1,1]$ and ...
2
votes
1answer
48 views

Finding all sequences of real numbers with a certain property

Let the series $\sum_{k \in N}a_k$ be convergent such that $\sum_{k \in N}a_k\neq 0$. Find all sequences $(b_k)_{k \in N}$ with the following properties: (i) the series $\sum_{k \in N}a_k b_k$ is ...
7
votes
6answers
470 views

Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?

Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the nth x = √(1st x √ 2nd x ... √nth x); Then $$\text{the ...