All Questions
Tagged with real-analysis summation
1,083
questions
3
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145
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Do the normal numbers form a Borel set?
Normal numbers have a 'random' expansion. For example, in base 10 it means that all digits $0,1,\dots,9$ occur 'equally often' in its decimal expansion. A longstanding open problem is: is $\pi$ a ...
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1
vote
1
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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
0
votes
1
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69
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Taking constant outside the summation
In the below picture they have taken $a$ outside? Is it allowed? Is both terms involving summation are equal?
My attempt
Term on left hand side is equal to $|\frac{1}{n}(a_1+... +a_n)-a|$
Whereas the ...
3
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0
answers
63
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Is there any function in which the Maclaurin series evaluates to having prime numbered powers and factorials? [duplicate]
I am searching for any information or analysis regarding the functions
$$f(x)=\sum_{n=1}^{\infty}\frac{x^{p\left(n\right)}}{\left(p\left(n\right)\right)!}$$
or
$$g(x)=\sum_{n=1}^{\infty}\frac{\left(-1\...
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3
votes
2
answers
296
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if $\lim\limits_{n \to \infty} b_n =0 $ then how to prove that $\lim\limits_{n \to \infty} \sum\limits_{k =1} ^n \frac{b_k}{n+1-k}=0$
in Problems in Mathematical Analysis I problem 2.3.16 a),
if $\lim\limits_{n \to \infty}a_n =a$, then find $\lim\limits_{n \to \infty} \sum\limits_{k=1 }^n \frac{a_k}{(n+1-k)(n+2-k)}$
The proof that ...
- 6,620
-1
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3
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170
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How do you find the value of $\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n^2}$? [closed]
Extra information which may be useful is that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ equals $\frac{\pi^2}{6}$ (Euler's solution to the Basel Problem).
4
votes
1
answer
659
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Power series question.
How would I go about solving $$\sum_{n=1}^{\infty} \frac{(n-1)x^{n-1}}{(n-1)!}$$ So far I have tried to of course consider the exponential power series, but I seem to get negative factorials.
- 89
3
votes
1
answer
239
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Is it possible to bound this sum?
Maybe this question is too simple, but I have been thinking about it for several days and I can't find the solution.
Is it possible to show that the there exists a universal constant such that the ...
- 433
0
votes
1
answer
81
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Prove that $\int_0^1\lfloor nx\rfloor^2 dx = \frac{1}{n}\sum_{k=1}^{n-1} k^2$
First of all apologies for the typo I made in an earlier question, I decided to delete that post and reformulate it
I am asked to prove that
$$\int_{(0,1)} \lfloor nx\rfloor^2\,\mathrm{d}x =\frac{1}{n}...
- 131
1
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0
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Does this series converge ? $\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$ Hint: $\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \ln{a}$
Does this series converge ?
$$\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$$
Hint: use the fact that
$$\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \lim_{h \to 0} \frac{a^h-1}{h} = \ln{a}$$
...
- 1,135
1
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1
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86
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Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
I first tried to use the Root Test.
$L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \...
- 1,135
0
votes
2
answers
69
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Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
Sorry for my last duplicate question. But for this question here, I did not find the same question
Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
$L = \lim_{n \to \infty} |\...
- 1,135
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0
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14
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Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$ [duplicate]
Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$
My attempt:
$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ with $a_n = n!/n^n$. This gives
$L = \lim_{n \to \infty} |(n+1)\frac{n^...
- 1,135
1
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0
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Prove for a sequence ($a_n$)$_n$ with converging partial sum $s_n = \sum_{k=1}^{n}a_k$ it holds that for a bounded, monotonically decreasing sequence
Prove that for a sequence ($a_n$)$_n$
with converging partial sum
$$s_n = \sum_{k=1}^{n}a_k$$ , it holds that for a bounded, monotonically decreasing sequence ($c_n$)$_n$, the series
$$\sum_{n=1}^{\...
- 1,135
1
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1
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Compute $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$
Here is a question in calculus. Compute the limit of the sequence: $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$?
There are in general three ways to compute ...
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