All Questions
1,809
questions
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3
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Evaluate $\prod_{n\geq2}\left(\frac{4}{e^2}\left(1+\frac{1}{n}\right)^{2n+1}\frac{n^2-1}{4n^2-1}\right)$
$$\prod_{n\geq2}\left(\frac{4}{e^2}\left(1+\frac{1}{n}\right)^{2n+1}\frac{n^2-1}{4n^2-1}\right)$$
I expand $\frac{n^2-1}{4n^2-1}$ as $\frac{(n-1)(n+1)}{(2n-1)(2n+1)}$. Then, $\left(1+\frac{1}{n}\...
user1254447
3
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2
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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 ...
- 6,563
2
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1
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Find the interval of convergence $\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$
Find the interval of convergence A,of the $$\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$$
and calculate the sum for each value of $x\in A$ .
My work
$$...
user1254447
1
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1
answer
101
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Compute ${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}$
Compute
$${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}.$$
What I have done so far
Lemma: $$\displaystyle{\mathop {\lim }\limits_{N \to \...
1
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0
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103
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Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
- 31
10
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3
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614
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Show that $\sum_{n=1}^{+\infty}\frac{1}{(n\cdot\sinh(n\pi))^2} = \frac{2}{3}\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{(2n-1)^2} - \frac{11\pi^2}{180}$
What I do so far
\begin{align*}
\text{Show that} \quad &\sum_{n=1}^{+\infty}\frac{1}{(n\cdot\sinh(n\pi))^2} = \frac{2}{3}\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{(2n-1)^2} - \frac{11\pi^2}{180} \\
\...
1
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3
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110
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Did I solve the problem $\lim_{n\to\infty}\sum_{k=0}^{n}\frac{1}{{n\choose k}} \to 2$ correctly?
I'm in a calculus class I need to prove that
$$\lim_{n\to\infty}\sum_{k=0}^{n}\frac{1}{{n\choose k}} \to 2$$
Is my below solution valid?
We are given the problem
$$\lim_{n\to\infty}\sum_{k=0}^{n}\...
- 133
0
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1
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37
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Let $s_n=\sum_l (e^{ix} − 1)^{-1}(e^{inx} − e^{-i(n-1)x})$ where $x=c+l\omega$ . How to show that $s_1+...+s_n=\sum_l \frac{1-\cos nx}{1-\cos x }?$
If i define $s_n=\sum_l (e^{i(c+l\cdot \omega)} − 1)^{-1}(e^{in(c+l\cdot \omega)} − e^{-i(n-1)(c+l \cdot \omega)})$ where $c \in [0,2\pi]$, how can i show that $$s_1+s_2+...+s_n=\sum_l \frac{1-\cos n(...
- 41
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1
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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
$$\int_{(0,1)} \lfloor nx\rfloor^2\,\mathrm{d}x =\frac{1}{n}...
- 131
3
votes
2
answers
206
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why does $\pi$ always show up in $\int_0 ^1 \frac{x^c}{1+x^k} dx$ if $c\neq mk-1$ for all $m \in \mathbb{N}$
when I posted this question I was interested in the sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{4n+3}$ but when I thought about the generalised sum $ \sum_{n=0} ^{\infty} \frac{(-1)^n}{kn+c +1}$ for all $...
- 6,563
3
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2
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161
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$\sin x \cos x$ defined as a Cauchy Product of their Taylor Series.
The goal of this project is to show that the Cauchy Product of two Taylor Series, $\sin x$ and $\cos x$, is equal to the Maclaurin Series of $\sin x\cos x$ . I am having trouble simplifying the ...
- 106
1
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1
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81
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Help needed in understanding how a certain result was found.
$$\sum_{j=1}^{a-1}\widetilde{\zeta}(j+1)\widetilde{\zeta}(2a-j)=\sum_{j=1}^{a-1}\widetilde{\zeta}(2j)\widetilde{\zeta}(2a+1-2j)$$
I'm trying to understand how the two are equal. At first, I thought ...
1
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1
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230
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Compute $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$
Here is a question in calculus. Compute the limit of the sequence: $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$?
There are in general three ways to compute ...
- 825
11
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3
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451
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How to evaluate $ \sum\limits_{k=0} ^{\infty} \frac{(-1)^k}{4k+3}$?
I was trying to solve the integral $\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx$ and I noticed I can do the following:
$$\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx=\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}} \...
- 6,563
5
votes
2
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185
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Generating Function $\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$
Closed Form For :
$$S=\sum_{k=1}^{\infty}\binom{2k}{k}^{-2}x^{k}$$
Using the Series Expansion for $\arcsin^2(x)$ one can arrive at :
$$\sum_{k=0}^{\infty}\binom{2k}{k}^{-1}x^{k}=\frac{4}{4-x}-4\arcsin\...
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