All Questions
Tagged with real-analysis summation
150
questions
213
votes
5
answers
45k
views
The sum of an uncountable number of positive numbers
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\...
- 5,910
39
votes
12
answers
90k
views
Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? [duplicate]
Can you please explain why
$$
\sum_{k=1}^{\infty} \dfrac{k}{2^k} =
\dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots =
2
$$
I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{...
- 433
25
votes
3
answers
6k
views
Does uncountable summation, with a finite sum, ever occur in mathematics?
Obviously, “most” of the terms must cancel out with opposite algebraic sign.
You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
- 4,470
36
votes
5
answers
9k
views
use of $\sum $ for uncountable indexing set
I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say
$$
\sum_{a \in A} a \quad \text{where A is ...
- 3,116
11
votes
2
answers
689
views
Infinite Series $\sum\limits_{k=1}^{\infty}\frac{k^n}{k!}$
How can I find the value of the sum $\sum_{k=1}^{\infty}\frac{k^n}{k!}$?
for example for $n=6$, we have
$$\sum_{k=1}^{\infty}\frac{k^6}{k!}=203e.$$
- 5,626
8
votes
5
answers
9k
views
Partial Sums of Geometric Series
This may be a simple question, but I was slightly confused. I was looking at the second line $S_n(x)=1-x^{n+1}/(1-x)$. I was confused how they derived this. I know the infinite sum of a geometric ...
user431865
21
votes
4
answers
6k
views
Can we add an uncountable number of positive elements, and can this sum be finite?
Can we add an uncountable number of positive elements, and can this sum be finite?
I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
user67803
11
votes
2
answers
560
views
Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
I have to prove that
$$\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\dfrac{1}{p!p}$$
How can I do that?
- 403
6
votes
5
answers
512
views
Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$
Was given the following infinite sum in class as a question, while we were talking about Taylor series expansions of $\ln(1+x)$ and $\arctan(x)$:
$$1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$$
...
- 61
10
votes
2
answers
888
views
Find $\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$
Find $$M:=\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$$
There's a solution here that uses complex numbers which I didn't understand and I was wondering if the following is also a correct method.
My ...
- 1,206
4
votes
4
answers
1k
views
Partial sum of the series $\sum\limits_{r=1}^{\infty} \frac{1}{r(r+1)}$
An Exercise from Apostol's Introduction to Analytic Number Theory which I am not able to solve.
Let $\mathsf{S_{n}}$ denote the $n$-th partial sum of the series: $$\sum\limits_{r=1}^{\infty} \frac{1}{...
anonymous
115
votes
25
answers
17k
views
Can an infinite sum of irrational numbers be rational?
Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear ...
- 1,841
0
votes
3
answers
149
views
Question about $a_n=\sum_{k=1}^n \frac{n}{n^2+k}\ $for $ n \in\mathbb{N}$
Please help me solve this. Please try to give some details also.
Let $a_n=\sum_{k=1}^n \frac{n}{n^2+k}\ $for $ n \in\mathbb{N}$ then the sequence $\{a_n\}$ is (choose the correct option):
...
- 2,223
61
votes
3
answers
3k
views
Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$
The equality$$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)\tag{1}$$follows from the fact that the sum of the first series ...
- 431k
13
votes
5
answers
1k
views
Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
I know how to prove that
$$\sum_1^{\infty} \frac{1}{n^2}<2$$ because
$$\sum_1^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}<2$$
But I wanted to prove it using only inequalities. Is there a way to do ...
- 6,756