All Questions
Tagged with real-analysis summation
1,083
questions
3
votes
0
answers
28
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What are these infinite sums of powers of integers, $n^p$, multiplying a quadratic in the Bessel function $J_n(nx)$ and its derivative $J'_n(nx)$?
What are explicit elementary functions of real $x$, for $0 < x < 1$, if they exist, for $p=1$ and $p=3$ of
$$\sum_{n=1}^\infty n^p [J_n(nx)]^2$$
$$\sum_{n=1}^\infty n^{p+1}J_n(nx)J'_n(nx)$$
$$\...
- 31
2
votes
4
answers
158
views
A problem on finding the limit of the sum
$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$
Show that, $\lim_{n\rightarrow\infty} u_n = 0$.
The only approach I can see is either ...
0
votes
1
answer
95
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Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$
I liked this problem because the result is a final answer, and ...
- 1,433
3
votes
2
answers
117
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Why does this proof work: $\sum\limits_{n=1}^ \infty \left(\frac{1}{4n-1} - \frac{1}{4n}\right)= \frac{\ln(64)- \pi}{8}$?
$$f(x):= \sum_{n=1}^ \infty \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f'(x) = \sum_{n=1}^ \infty ( x^{4n-2}- x^{4n-1})= \frac{x^2}{(1+x)(1+x^2)}$$
$$\int_0 ^1 \frac{x^2}{(1+x)(1+x^2)}=...
- 6,620
0
votes
4
answers
196
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How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \...
- 6,620
5
votes
1
answer
183
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How to rigorously prove that $\sum\limits_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n} )=\frac{\ln(64)- \pi}{8}$?
How to rigorously prove that $\sum\limits_{n=1}^ \infty\left( \frac{1}{4n-1} - \frac{1}{4n}\right) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \left(\frac{x^{4n-1}}{4n-1} - \frac{x^...
- 6,620
3
votes
1
answer
63
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Proving Existence of Sequence and Series Satisfying Hardy-Littlewood Convergence Condition Prove that for every $\vartheta, 0 < \vartheta < 1$,
Prove that for every $\vartheta, 0 < \vartheta < 1$, there exists a sequence $\lambda_{n}$ of positive integers and a series $\sum_{n=1}^{\infty} a_{n}$ such that:
\begin{align*}
(i) & \...
4
votes
1
answer
169
views
Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$
ORIGINAL QUESTION (UPDATED):
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
0
votes
2
answers
62
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Comparing integral with a sum
Show that \begin{equation}\sum_{m=1}^k\frac{1}{m}>\log k.\end{equation}
My intuition here is that the LHS looks a lot like $\int_1^k\frac{1}{x}\textrm{d}x$, and this evaluates to $\log k$. To ...
- 135
0
votes
0
answers
21
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Changing the order of the sum in a measure question [duplicate]
Let $(X,\mathcal{A})$ be a measurable space and $(\mu_n)$ be a sequence of measures which satisfies $\mu_n(X)=1$. I want to show that the function defined as
$$
\nu(E)=\sum_{n=1}^{\infty}2^{-n}\mu_n(E)...
- 53
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
111
views
For $a>1$ and a fixed $N$ does $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \left(1- \frac{a}{a+N +k} \right)$ converge?
I was trying to find a proof for Raabe's Ratio test: If $x_n$ is a positive sequence of real numbers, if $\lim\limits_{n \to \infty } n \left(\frac{x_n}{x_{n+1}}-1 \right) >1$ then $\sum\limits_{...
- 6,620
2
votes
1
answer
87
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Convergence of the series $f(a_n)a_n$
Let $f : \mathbb{R} \to \mathbb{R}$ a monoton function in $[−r, r]$ for some $r>0$. Prove that the if $$\sum a_n$$ converge absolutely then $$\sum f(a_n)a_n$$ converges absolutely
I have not idea ...
- 145
1
vote
1
answer
123
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Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
1
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0
answers
30
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Insufficient boundedness for integration via Darboux sums
I have the simple integral $$\int_0^1 \frac{dx}{\sqrt{1-x}},$$ which I would like to constrain between two Darboux sums whose summands have absolute values not all zero. My attempt in doing so begins ...
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