All Questions
29
questions
6
votes
1
answer
229
views
Is it always true that if $x_n\to0$, $y_n\to0$ there exist $\epsilon_n\in\{-1,1\}$ such that both $\sum\epsilon_nx_n$and $\sum\epsilon_ny_n$ converge? [duplicate]
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \...
- 6,620
2
votes
1
answer
100
views
How to prove that for fixed number $m$ positive numbers the sequence $\frac{\sum\limits_{k=1}^ma_k^{n+1}}{\sum\limits_{k=1}^ m a_k^{n}}$ is monotone?
I saw this question in my book
Let $a_1 , a_2 , \dots,a_m$ be fixed positive numbers and $S_n =\frac{\sum \limits_{k=1}^ m a_k^{n}}{m}$ Prove that $\sqrt[n] {S_n} $ is monotone increasing sequence
...
- 6,620
2
votes
1
answer
220
views
Proving a property related to $M/M/c$ queues - Queueing theory.
My goal is to show that in a $M/M/c$ queueing system it is satisfied that
$$ L_s = L_q + \frac{\lambda}{\mu}, $$
where $L_s$ represents the average number of costumers in the system, $L_q$ represents ...
- 1,141
1
vote
1
answer
91
views
Evaluating a finite sum.
Amid one exercise I was solving, I came across the following finite sum:
$$ \sum_{n=0}^{N} n\left(\frac{3}{2}\right)^n.$$
This sum was evaluated in one of my classes, but I don't understand/agree with ...
- 1,141
1
vote
0
answers
67
views
Proof of Cesàro summation
This is a proof I came up with while working on the textbook Understanding Analysis:
Supposing $x_{n} \rightarrow x$, we have that
$$s_n = \frac{1}{n} \sum_{k=1}^{n} x_k \rightarrow x$$
Let $\epsilon \...
- 402
1
vote
0
answers
30
views
Prove $\sum_{n=1}^N a_n b_n = a_N B_N - \sum_{n=1}^{N-1}(a_{n+1} - a_n) B_n$
Prove summation by parts:
$$
\sum_{n=1}^N a_n b_n = a_N B_N - \sum_{n=1}^{N-1}(a_{n+1} - a_n) B_n
$$
where $B_N$ means $\sum_{n=1}^N b_n$ and $B_0 = 0$.
The proof is via induction. The base case is ...
- 4,450
-1
votes
3
answers
97
views
Proving an infinite sum can take on arbitrarily small values
Show that we can pick $x>0$ sufficiently small that \begin{equation}\frac{x}{2\cdot 3}+\frac{x^2}{3\cdot 4}+\cdots+\frac{x^n}{(n+1)(n+2)}+\cdots<k\end{equation} for any $k\in(0,1).$
My idea: ...
user1022331
1
vote
2
answers
83
views
Find all the integers which are of form $\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}, a,b,c\in \mathbb{N}$, any two of $a,b,c$ are relatively prime.
I have a question which askes to find all the integers which can be
expressed as
$\displaystyle \tag*{} \dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}$
where $a,b,c\in \mathbb{N} $ and any two of $a,b,...
- 921
3
votes
1
answer
55
views
Calculating a sum $\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$
I want to calculate this sum, while $0<p<1$:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$$
Is this correct:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}=\frac{1}{1-p}\cdot \frac{1-\frac{1}{(1-p)^k}}{1-\frac{1}{1-...
- 627
3
votes
1
answer
237
views
Proof verification: if $a_n, b_n>0$ and $\lim\limits_{n \to\infty} \frac{a_n}{b_n}=L_1$ with $L_1>0$, then if $\sum a_n$ converges, so does $\sum b_n$
I'm trying a proof technique I'm not used to for limits on fractions, which attempts to avoid an epsilon-delta approach similarly to how the single variable chain rule is proved in baby Rudin, and I ...
- 59
1
vote
1
answer
36
views
How to rigorously transform a sum of tuples into sum of its components?
Consider a function $f(x)$, which maps to real numbers.
Let $x \in \mathcal{S} = \{(0,0),(0,1), (1,0), (1,1)\}$.
I want to define a quantity $I = \sum_{x \in \mathcal{S}} f(x)$.
Now, let $x_1$ be the ...
- 1,156
1
vote
1
answer
70
views
Summation of function subtraction with a limit as $\epsilon \rightarrow 0$
I got stuck in a problem in the middle of my calculations of integrals and sums.
$$\lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty f(n-\epsilon)-f(n+\epsilon)=0$$
where $f$ is continuous on all of the ...
- 59
2
votes
1
answer
70
views
Test the convergence of $\sum_{n=1}^\infty \frac{n^{1/n} - 1}{n}$ [duplicate]
Test the convergence of $$\sum_{n=1}^\infty \frac{n^{1/n} - 1}{n}$$
My Attempt: Using the root or ratio test would be too inconvenient here. Looking at the denominator, I used the Cauchy Condensation ...
- 4,335
2
votes
2
answers
63
views
Show this sum is uniformly bounded in $N$ and in $i$.
For $N>0, \ d>0$, I am considering $N$ points $(y_1,...,y_N)$ in $(\mathbb{R}^d)^N$ such as there exists a constant $c>0$ with :
$$\underset{1 \leq i,j \leq N \atop i \neq j}{\min} |y_i-y_j| \...
- 2,190
0
votes
1
answer
49
views
Convergence radius of $\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$
Find the convergence radius of $$\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$$
I've recently started self-learning about series, so I'm having a little trouble. Looking at this example, I tried ...
- 11.6k
1
vote
0
answers
81
views
Prove that $1 < \sum\limits_{n = 1001}^{3001} \frac 1 n < \frac 3 2.$
Let $x = \sum\limits_{n = 1001}^{3001} \frac 1 n.$ Prove that $1 < x < \frac 3 2.$
My attempt $:$ What I did is as follows
By AM-HM inequality we have
\begin{align*} x =\sum\limits_{1001}^{3001}...
- 2,912
0
votes
1
answer
46
views
Sum of products basics (proof verification)
I know that for a collection of random variables $\{x_i\}_{i=1}^n$, I have
$$\sum_{i,j=1}^nE(x_ix_j)=\sum_{i=1}^{n}E(x_i^2)+2\sum_{i<j}E(x_ix_j). \quad (a)$$
Then triangle inequality gives
$$\Big\...
- 2,689
1
vote
2
answers
136
views
Computation of: $\lim_{n\to\infty}\left(\ln\left(1+\frac{1}{n^2+1}\right)^n+\ldots+\ln\left(1+\frac{1}{n^2+n}\right)^n\right)$
Evaluate:
$$\lim_{n\to\infty}\left(\ln\left(1+\frac{1}{n^2+1}\right)^n+\ldots+\ln\left(1+\frac{1}{n^2+n}\right)^n\right)\;n\in\mathbb N$$
My attempt:
Using the manual limit:
$$\lim_{x\to 0}\frac{\...
- 4,670
0
votes
1
answer
69
views
Snails and Sums
At the beginning of a $10\,\mathrm m$ long rubber band sits a snail.
Every day it crawls one meter ahead. Every night, when the snail is resting, an evil man stretches the tape evenly by $10\,\mathrm ...
- 2,482
1
vote
1
answer
219
views
Baby Rudin ex. 3.8 proof verification
The question asks: if $\sum a_n$ converges, $\{b_n\}$ is monotonic and bounded, prove that $\sum a_nb_n$ converges.
My proof goes as follows:
Let $\varepsilon>0$, and let $S_k$ denote $k$-th ...
- 13
10
votes
2
answers
889
views
Find $\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$
Find $$M:=\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$$
There's a solution here that uses complex numbers which I didn't understand and I was wondering if the following is also a correct method.
My ...
- 1,206
0
votes
1
answer
36
views
Given $\sum^\infty_{k=1}\frac{(-1)^{k+1}}{\sqrt{k}}$ converges: [closed]
Assume $\sum^\infty_{k=1}a_k$ and $\sum^\infty_{k=1}b_k$ are both non-absolutely convergent. Find specific examples of $\sum^\infty_{k=1}a_k$ and $\sum^\infty_{k=1}b_k$ such that $\sum^\infty_{k=1}(...
- 13
2
votes
2
answers
190
views
The series $\sum_n^\infty a_n^p$ where $\{a_n\}_{n=1}^\infty$ is a convergent, strictly positive sequence
Suppose that $\{a_n\}_{n=1}^\infty$ is a sequence of strictly positive numbers and that $\sum_n^\infty a_n=A$ is a convergent series. Suppose that $p >1$.What can you say about the series $\sum_n^\...
- 3,425
1
vote
9
answers
172
views
How to prove that $e^k > k$?
I'm working on series and I'm always stuck at one point where I dont know how to prove that $$e^k > k$$
I'm trying to show by a comparaison test that
$$\sum_{n=1}^\infty \frac{n}{e^n}$$ ...
- 613
0
votes
3
answers
91
views
Prove through induction that $\sum_{a=1}^{b}a(b-a)=\frac{b(b-1)(b+6)}{6}$ [closed]
Given that $\sum_{a=1}^{b}a=\frac{a(a+1)}{6}$ prove through induction that $$\sum_{a=1}^{b}a(b-a)=\frac{b(b-1)(b+6)}{6}$$
Normally I would start by showing that this statement is true for $b=1$ and ...
- 1,928
0
votes
1
answer
25
views
Sum Manipulation to be less than a constant
I'm stuck with some basic sum manipulation in the middle of a proof. We have that:
All terms are non-negative.
$\sum\limits_{i=1}^\infty x_i=1$ (and the same goes for $y_i).$
Is there a way to show $...
user510652
2
votes
0
answers
67
views
$\sum_{n=1}^{\infty}a_n=\sum_{n\in\mathbb{N}}a_n$
I want to prove that $\sum_{n=1}^{\infty}a_n=\sum_{n\in\mathbb{N}}a_n$ if $a_n$ is non-negative. Call the first series $A$ and the second series $B$. I'm just assuming that the definition of the ...
- 3,037
3
votes
2
answers
163
views
Does this series converge conditionally $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$
$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$
According to my understanding, if $\sum\left|a_n\right|$ diverges but $\sum a_n$ converges, then the series is conditionally convergent.
For $\...
- 329
1
vote
1
answer
27
views
Recursion solution doesn't seem correct
I'm studying real analysis at the moment on my own. So I don't have a professor to ask if I'm not sure about my answer to an exercise from my text. So I'll ask you guys.
The question is
Let $d$ ...
- 13