All Questions
Tagged with real-analysis summation
1,083
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
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
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
58
votes
7
answers
25k
views
Is it possible to write a sum as an integral to solve it?
I was wondering, for example,
Can:
$$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$
Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals.
But ...
- 11.2k
42
votes
6
answers
7k
views
If $f(x)=\frac{1}{x^2+x+1}$, how to find $f^{(36)} (0)$?
If $f(x)=\frac{1}{x^2+x+1}$, find $f^{(36)} (0)$.
So far I have tried letting $a=x^2+x+1$ and then finding the first several derivatives to see if some terms would disappear because the third ...
- 621
39
votes
12
answers
90k
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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
37
votes
3
answers
3k
views
An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$
$n$ is a positive integer, then
$$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$
please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$.
I want to find a ...
- 7,819
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
$$
\sum_{a \in A} a \quad \text{where A is ...
- 3,116
34
votes
3
answers
6k
views
Why does Wolframalpha think that this sum converges?
Looking at the sum:
$$\sum_{n=1}^\infty\tan\left(\frac\pi{2^n}\right)$$
I'd say that it does not converge, because for $n=1$ the tangent $\tan\left(\frac\pi 2\right)$ should be undefined. But ...
- 555
32
votes
1
answer
1k
views
Geometric representation of Euler-Maclaurin Summation Formula
In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
- 7,093
31
votes
1
answer
1k
views
Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$
For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that
$$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$
The case without $\...
- 5,217
29
votes
2
answers
829
views
How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$
How can we prove the following
$$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$
I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} \...
user208998
29
votes
1
answer
570
views
A nice Combinatorial Identity
I am trying to show that $\forall N\in\mathbb{N}$,
$$\sum\limits_{n=0}^{N}\sum\limits_{k=0}^{N}\frac{\left(-1\right)^{n+k}}{n+k+1}{N\choose n}{N\choose k}{N+n\choose n}{N+k\choose k}=\frac{1}{2N+1}$$
...
- 1,984
28
votes
4
answers
23k
views
Summation Symbol: Changing the Order
I have some questions regarding the order of the summation signs (I have tried things out and also read the wikipedia page, nevertheless some questions remained unanswered):
Original 1. wikipedia ...
- 1,053
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