All Questions
Tagged with summation power-series
362
questions
1
vote
1
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49
views
Show that $F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$
I've been working on a recent exercise question where I was asked to show that:
$$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is ...
- 115
1
vote
0
answers
36
views
Field of convergence, prove it's open?
For the following power series $$ \sum_{n=1}^{ \infty} \bigg( 1 + \frac{(-1)^n}{n} \bigg)^{ n^2} \frac{(2x+1)^n}{n} $$ I proved that the radius of convergence is $r= \frac{1-e}{2 \ e} $.
How may I ...
- 421
0
votes
3
answers
31
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an infinite series that i couldn't figured out how to sum up
I have the series which is $$\sum_{n=0}^{\infty} 2^{-n(x-1)}$$ and from the ratio test it converges for all $x\geq 2$ but how can i find the general sum of the series wrt $x$
- 114
1
vote
1
answer
540
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Calculating sum of series using derivative of a function
We're given the following problem:
"We know that $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k $ for $ -1 < x < 1 $. Using the derivative with respect to $x$, calculate the sum of the following ...
- 181
9
votes
1
answer
1k
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Proving $\pi=(27S-36)/(8\sqrt{3})$, where $S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$ [closed]
I have to prove that:
$$\pi=\frac{27S-36}{8\sqrt{3}}$$
where I know that $$S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$$
Where do I get started?
- 125
0
votes
1
answer
65
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Computing the sum of an alternating power series
I'm asked to find the sum of a power series
for all x in the interior of the domain of convergence, which I found to be [-1,1]. The question also gives the hint to take the second derivative of the ...
- 41
0
votes
2
answers
62
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Show that a the sum function of a power series is differentiable twice and that $f''(x) = \frac{1}{49-7x}$
I am studying for my analysis exam and are to consider the power series
$$
\sum_{n=2}^\infty \frac{1}{n(n-1)7^n}z^n
$$
with the sum function for $x \in ]-7,7[$ given by
$$
f(x) = \sum_{n=2}^\infty \...
- 917
0
votes
2
answers
69
views
Infinite sum power series
I would like to show
$$
\sum_{r=0}^{\infty}\frac{1}{6^r} \binom{2r}{r}= \sqrt{3}
$$
I have tried proving this using telescoping sum, limit of a sum, and some combinatorial properties but I couldn't ...
- 41
1
vote
2
answers
461
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Sum of finite series using partial fraction
I'm quite stuck with the following problem. I have seen on this forum that there is already an answer for the infinite sum to the problem but I can't seem to find how to find the sum for a finite ...
- 181
0
votes
1
answer
49
views
Theoretical Procedure for Power Series Equation:
If I have the following equation: \begin{equation}2c_0(x-1)+\sum_{k=2}^\infty[(c_{k-2}+2c_{k-1})(x-1)^k]+\sum_{k=0}^\infty[(c_{k+2}(k+2)(k+1)+kc_k+(k+1)c_{k+1}+c_k)(x-1)^k]=0 \end{equation}
I was ...
- 2,653
0
votes
1
answer
26
views
Given the following summation is there a way to combine given the following orientation?
I have the following summation: \begin{equation}\sum_{n=2}^\infty c_n(n)(n-1)(x-1)^{n-2}+(x+1)\sum_{n=1}^\infty nc_n(x-1)^{n-1}+\sum_{n=1}^\infty nc_n(x-1)^{n-1}+(x+1)^2\sum_{n=0}^\infty c_n(x-1)^n +2(...
- 2,653
0
votes
0
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33
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Convergence of specific power series
I have to evaluate pointwise/uniform/total convergence of this series and I didn't quite understand how to do it.
$$\sum_{k=2}^{+\infty}{\ln k \over 2+\sin k}x^k$$
For pointwise convergence: it ...
- 415
1
vote
2
answers
433
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sigma notation- squaring the entire sum
Could someone please tell me how to expand this?
$\bigg(\sum_{i=1}^ne^{at_i-\frac{1}{2}\sigma^2t_i+\beta t_i}\bigg)^2$
i know the general formula goes something like this:
$\bigg(\sum_{i=1}^na_i\...
2
votes
1
answer
132
views
Upper bound for the sum of non-integer powers
Let $a_1, a_2, \ldots, a_k$ be a positive integers such that $a_1 + a_2 + \cdots + a_k = K$.
Is it possible to find an upper bound such that $$a_1^p + a_2^p + \cdots+ a_k^p \le f(K)$$ where $0 < p ...
- 23
1
vote
0
answers
17
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Find the power series from the given maclaurin sequence
I have made a post regard this particular question but was incorrect in what I was asking.
The sequence needs to be written in sigma notation, not as a summation.
The given sequence is:
$$x+2x^3+x^3+...
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