For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

learn more… | top users | synonyms (5)

2
votes
1answer
34 views

If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

Let $a_{n} \geq 0$ and $b_{n}>0$ for each $n$ in $\mathbb{N}$ and suppose that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} ...
1
vote
1answer
72 views

Part of proof of term-by-term integration

I want to prove the theorem of term-by-term integration for lebesgue integrable functions (denoted as $L^1$ functions): Suppose $(g_n)$ is a sequence of $L^1$ functions over a measure space $(X,\sigma ...
0
votes
0answers
9 views

Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations. Let $A$ be an ...
0
votes
1answer
22 views

How to shift the weighted mean of a monotically increasing series of values

I have a monotonically increasing series of values $X, x\in[0,1] \ \forall i\in1,2,...,60$ the weighted mean of which is defined by $$\bar{x}=\frac{x_1+\sum_{i=2}^{60} i(x_i-x_{i-1})}{x_{60}}.$$ Given ...
5
votes
2answers
137 views
+100

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ ...
2
votes
1answer
30 views

Is there any known way to sum a subserie (square indices) of geometric series?

I was interested in the following sum. Although im not sure there exist any known way to sum this...it seems rather difficult. Can we sum for $0<r<1$ $$\sum_{k=0}^{\infty}r^{k^2}= ...
-1
votes
1answer
22 views

Find the bases of the vector space of terminal sequences

Let V be the vector space of the sequences $ a = (a_0 , a_1 , a_2 , ...) $ of real numbers who are terminally - finally zero sequences (There is $ N $ such that $ a_n = 0 $ for every $ n > N $ ). ...
3
votes
2answers
67 views

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to:

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to: I do not know what to try to find the solution. A hint along with the explanation ...
0
votes
1answer
65 views

How to calculate $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$

Problem Show that $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$ by the following consequence. Consider the sequence $(s_n)$ defined by $s_n=1 + 1/2 + \cdots + 1/n$. ...
-4
votes
0answers
32 views

How much would this child get in the 30 days of the month of June.

This is the 10th question in my assignment. I'm not too sure if I got the common difference for this question correct. Here's the question: A child tries to negotiate a new deal for her pocket money ...
2
votes
3answers
47 views

The sum of the 2nd and 3rd term is 12, and the sum of the 3rd term and 4th term is 60.

This is the 9th question in my assignment. It's another confusing question. Here's the question: In a geometric progression, the sum of the 2nd and 3rd terms is 12, and the sum of the 3rd and 4th term ...
1
vote
1answer
33 views

How far does does this guy walk if he empties 24 barrow fulls and returns to the load each time?

This is the 6th question in my assignment. It sounds quite confusing to me. Here's the question: A man moves a load of soil for top dressing an orchard by emptying barrow loads in a line 20 meters ...
0
votes
3answers
53 views

If $x=\frac{2ab}{a+b}$ show that $1/a, 1/x, 1/b$ is an arithmetic progression.

This is the 5th question in my assignment. It's a really freaky looking problem. I haven't come up with an answer yet but I made an attempt with the working out. Here's the question: If ...
1
vote
1answer
31 views

The 11th term and the 12th term have the same sum as the first 10 terms of the same arithmetic progression.

This question is part of my advanced maths assignment on arithmetic and geometric progressions. Here's the question: Q4. Given that both the sum of the first ten terms of an arithmetic progression and ...
0
votes
1answer
19 views

Find out the value of x for which the given series convergent?

Given series $\sum_{n=1}^{\infty} 2^n (\tan x)^{n^2}$ Find out the real numbers for which the following integral is convergent? For solution i take $a_n=2^n (\tan x)^{n^2}$ and apply Cauchy root ...
0
votes
0answers
22 views

Evaluation of r^3 from zero to N (Sigma Notation)

I have to evaluate the following expression, $\sum_{n = 0}^{9} (n^3 -1)$. I know that $\sum_{n = 1}^{9}(n^3 -1)$ is given by $\frac{1}{4}(9)^2(9+1)^2 - 9$ But, how do I do this from zero to ...
2
votes
1answer
99 views

How can we see that $ \sum_{n=0}^{\infty}\frac{2^n(1-n)^3}{(n+1)(2n+1){2n \choose n}}=(\pi-1)(\pi-3) $?

I wonder will it help me so prove it if I was to decompose it into partial fractions? Mathematica approves of the identity; it is converges. can anyone help me to prove it? $$ ...
2
votes
2answers
66 views

Does sum of sequence $\sum \frac{1}{n (1+\frac{1}{2}+ \dots + \frac{1}{n})}$ converge?

A direct proof is hard, and I tried to develop the following equivalence: The convergence of $\sum \frac{x_n}{x_1+x_2+\dots+x_n}$ is the same as $\sum x_n$, but I still don't know how to prove it
1
vote
3answers
145 views

A Chinese Exam Question which is…quite hard

Let $f(x)=x^2-2x-3$, and $x_n$ be some sequence. $x_1=2$, $x_n =$ the $x$ coordinate of the point of intersection of the $x$ axis and the line joining $P(4,5)$ and $Q_n(x_n, f(x_n))$. Find an ...
1
vote
3answers
47 views

Where does the following series converge? [closed]

Using integrals or by any other method find: $\lim_{n \rightarrow\infty} \sum_{i=1}^{n}\frac{1}{n+i}$
3
votes
1answer
26 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j ...
1
vote
1answer
30 views

Series with recursive terms: reciprocal of the sums of previous terms

What are some references and material that are devoted to studying series of the form $x_1+\frac1{x_1}+\frac{1}{x_1+\frac1{x_1}}+\dotsc$ where $x_1>0$ is given? For instance, my (non math) friend ...
2
votes
2answers
69 views

What is the sum of this series: $1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 +\cdots$?

Say I have a series like the following; $$1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 + \frac{1 \times 6 \times 11}{5 \times 10 \times 15}x^3 + \cdots.$$ How do I find the sum of this? ...
10
votes
5answers
300 views

Number of points of accumulation of a sequence

Can a sequence have infinitely many points of accumulation i.e. we can extract infinitely many subsequences from it s.t. they all converge to their respective point of accumulation? I have the ...
-2
votes
1answer
14 views

Averge of sequence converges, sequence bounded? [closed]

Suppose $\frac{\sum_{k=1}^{n} |a_n|}{n}\leq M$ for every $n\in \mathbb N$,does it imply that $\sup |a_n|<\infty$ ? Can someone give me a hint?
1
vote
2answers
45 views

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$?

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$? where $_{0}F_{1}$ is the hypergeometric series?
2
votes
4answers
112 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
1
vote
2answers
42 views

Prove the monotone convergence theorem for sequences of Lebesgue-integrable functions

I'm trying to prove the monotone convergence theorem for $L^1$ functions: Suppose $(f_n)$ is a sequence of $L^1$-functions (i.e Lebesgue-integrable functions) over a measure space $(X,\sigma (X), ...
1
vote
0answers
39 views

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here's Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} ...
1
vote
3answers
32 views

Searching for a sequence of functions

Consider the following set of functions: $$ A=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, \lim_{r\searrow 0}\frac{f(r)}{r}\text{ exists}\right\}. $$ Is there a sequence $(f_n)\in A^{\mathbb{N}}$ such ...
2
votes
1answer
90 views

How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$? [duplicate]

while doing the Integration problem using Limit of a sum approach i have a doubt how $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln2$$ by infinite geometric series we have ...
0
votes
3answers
30 views

Proving the set of subsequences of a sequence are uncountable

I am attempting to solve the following problem. Let ($s_n$) be a subsequence of real numbers. Prove that the set of subsequences of ($s_n$) is uncountable. I was thinking that approaching this ...
3
votes
1answer
74 views

Infinite product include summation

I would like to find an infinite product of$$\prod _{n=2}^{\infty} \left(1+\frac{(-1)^{n-1}}{a_n}\right)$$ where $a_n = \sum_{k=1}^{n-1} \frac{n!(-1)^{k-1}}{k!} $ I tried to compute $a_2 , a_3 ...
7
votes
0answers
77 views

Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
2
votes
1answer
67 views

If $|z_n-z_m|> 2$ for every $n\ne m$ then $\sum \frac{1}{z_n^3}$ converges

Let $(z_n)$ be a sequence of non-zero complex numbers such that $\forall n,m, n\neq m\implies |z_n-z_m|> 2$ Prove that $\sum \frac{1}{z_n^3}$ converges. I'm clueless with this problem. A ...
1
vote
0answers
54 views

The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
6
votes
3answers
124 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
0
votes
2answers
42 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
2
votes
0answers
38 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
0
votes
2answers
19 views

bounded but not convergent sequences

I am not sure that if this question has a positive answer...I am looking for a sequence of real numbers $(p_{n})_{n\geq 1}$ such that $-1<\lim _{n}\inf p_{n}\leq \lim_{n}\sup p_{n} <1$ (as ...
0
votes
1answer
27 views

Series proof needed

I have following equations but I do not know the proof. Kindly provide the proof or give me some reference to look into. Here are the equations. 1- ...
1
vote
1answer
25 views

What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
-2
votes
0answers
27 views

Why is it that when n ≥ 1 the series is $\le$ 1/4 [closed]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
0
votes
0answers
21 views

show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
2
votes
1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
votes
0answers
40 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
0
votes
1answer
21 views

Sum of product of convergent series

Suppose $\{a_n\},\ \{c_n\}\subset\mathbb{R}$ satisfies $\lim\limits_{n\to\infty} a_n = a\in \mathbb{R}$, $\lim\limits_{n\to\infty} c_n = 1$. Prove that $$ \lim\limits_{M\to\infty} ...
0
votes
1answer
96 views

Is there any summation method that assigns $ \sum_{n=1}^\infty \frac{1}{n} =-\frac{\pi}{2}$

I don't know too much about alternate summation methods, but am interesting to know if any give the sum of the harmonic series to be $$-\frac{\pi}{2}$$
0
votes
2answers
53 views

$\sum_{k=1}^n 1/k - \log n$

I got this question : $$a_n = \sum_{k=1}^n \frac 1k - \log n$$ I proved that $\lim a_n $ exist. Now I have to prove: $$ 0<a_n-\lim a_n\le \frac 1n $$ for every $n \in \mathbb N$. I tried ...
0
votes
1answer
43 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...