All Questions
Tagged with real-analysis summation
1,083
questions
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
5
votes
2
answers
142
views
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, $n(r) = \#\{\alpha_j \le r\}$. Prove $\int_0^1n(r)dr = \sum_{j=1}^\infty(1-|\alpha_j|)$.
I'm trying to solve the following exercise from chapter 15 of Rudin's Real and Complex Analysis:
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, and let $n(r)$ be the number of terms in the ...
- 5,487
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
5
votes
1
answer
256
views
Sum the infinite series of $\frac{1}{r^3+1}$
Is there a definite value for the sum:
$S=\displaystyle\sum_{r=1}^{\infty} \frac{1}{r^3+1}$
And if so, how would I arrive at finding this sum?
I have tried reducing the above into partial fractions,...
- 611
17
votes
1
answer
574
views
Prove that $\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$
I want to prove that :
$\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$ when $n$ tends to $+\infty$
Perhaps $\sum_{k=0}^n{e^{ik^2}}$ is bounded, I don't know.
Do you have ideas ?
- 572
1
vote
0
answers
83
views
Analogue for finite sums of $\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g)$
Please help me to find an analogue for finite sums of
$$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) \tag{*}$$
where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
- 11
13
votes
1
answer
692
views
Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$
I am having trouble proving that
$$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$
I know that
$$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty \frac{(2x)^{2n}}{...
- 11.6k
0
votes
1
answer
101
views
Generalizing $\sum_{n=1}^{N}f(n)=\int_{a}^{b}f(x)dx+\int_{a}^{b}f'(x)\langle x\rangle dx+f(a)\langle a\rangle-f(b)\langle b\rangle$
This might be an insanely dumb question but I just spent about four hours figuring out how to show that for $0<a<1$ and $N\leq b<N+1$, the following holds
$$\sum_{n=1}^{N}f(n)=\int_{a}^{b}f(...
- 5,619
10
votes
2
answers
818
views
Does this double series converge?
$$\sum\limits_{y=1}^{Y}\sum\limits_{z=1}^{y} a^{y-1} b^y \binom{y-1}{z-1} (c + 2z)^d $$
Does this series converge when $Y=∞$? If the series converges, what does it converge to? If the series does not ...
- 113
1
vote
1
answer
145
views
Monotonicity of sequence determined by a summation
I'm studying for my calculus exam and I'm stuck with the next exercise, I have to find out the monotonicity of the sequence $\{a_n\}$ with the next general term:
$$a_n=\sum \limits_{p=1}^n\frac{1}{n+...
- 703
0
votes
3
answers
43
views
Is $\sum_{k=m+1}^{n} \frac{1}{2^k} < \frac{1}{2^m} $ true in general?
Is
$ \sum_{k=m+1}^{n} \frac{1}{2^k}
< \frac{1}{2^m}
$
true in general? Does it require induction on both $m$ and $n$?
- 1,725
8
votes
3
answers
398
views
Evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2}$ [duplicate]
Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
- 6,510
1
vote
1
answer
499
views
Properties of Sums
I am proving an integral property. Is the following manipulation valid in sums?
$\sum\limits_{i=1}^n x_i = \sum\limits_{i=1}^n y_i$
Then $\sum\limits_{i=1}^n x_i\cdot p_i = \sum\limits_{i=1}^n y_i\...
- 7,418
36
votes
5
answers
9k
views
use of $\sum $ for uncountable indexing set
I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say
$$
\sum_{a \in A} a \quad \text{where A is ...
- 3,116
7
votes
1
answer
350
views
Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$
Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges.
I see that,
$ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} \frac{1}{n+\frac{...
- 125