All Questions
Tagged with summation power-series
362
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If $f(x) = \sum_{n=1}^\infty \frac{2^n}{n}x^n$ for $x \in ]-\frac{1}{2}, \frac{1}{2}[$ then what is $f'(x)$?
If $$f(x) = \sum_{n=1}^\infty \frac{2^n}{n}x^n \: \: \text{for}\: \: x \in ]-\frac{1}{2}, \frac{1}{2}[$$ then what is $f'(x)$?
Attempt
It turns out that $\rho = \frac{1}{2}$ is the radius of ...
- 539
1
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0
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56
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How to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})\;$ as a linear function?
Suppose we have a finite set $I \subseteq \mathbb{N}$, and $\alpha_i\in [0, 1]$ are fixed numbers for all $i\in I$.
Is there a way to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})...
- 342
-1
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1
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68
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A closed form for the sum of a series $\sum_{n=1}^{\infty}x^{n\alpha} /\Gamma{(n \alpha)}$
Let $\alpha \in (0,1)$. Is there a closed form for the sum $\sum_{n=1}^{\infty}x^{n\alpha}/\Gamma{(n \alpha)} $ ?
- 3,165
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105
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Manipulation with the following infinite sum
Calculating some observable, I obtained the following-like converges sum
$$
S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p),
$$
where $F$ - some ...
- 121
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votes
2
answers
119
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How do I find the partial sum of the Maclaurin series for $e^x$?
In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$.
Wolfram | ...
- 3,515
1
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1
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151
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Is it possible to express a power series with squared coefficients as a function of the series without squared coefficients?
Suppose I have two sums, $P(x)$ and $Q(x)$:
$$P(x)\equiv \sum_{n=0}^N a_n x^n$$
$$Q(x)\equiv \sum_{n=0}^N a_n^2 x^n$$
Is there a way to express $Q(x)$ as a function of $P(x)$?
Context: I have a ...
- 111
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59
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Interchange of differentiation and summation in infinite sums
Is it possible to interchange differentiation and summation for infinite but also uniformly convergent sums, like:
$\dfrac{d}{dx} \sin{x} = \dfrac{d}{dx}\sum_{n=0}^\infty (-1)^{n} \, \dfrac{x^{2n + 1}}...
- 181
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67
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Is it possible to rewrite this sum in terms of some power series?
Is it possible to rewrite this sum in terms of some power series, maybe some cosine power series?
$$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
- 181
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Finding the nth sum of a series [closed]
I am to find the sum of a series that takes this format
$
\sum_{i=1}^{n}\frac{1}{i^\beta}
$
$
\beta
$
is a positive real number
How to approach the partial sum of the above series and can obtain its ...
- 3
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1
answer
59
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Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Let $(a_i), (b_i)$ be two non-negative sequence.
Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Does it necessarily mean that ...
- 5,473
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59
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$\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$
I am stuck in finding the sum of $\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$.
The sum looks quite similar to a negative binomial sum but I can't really find the exact form. Can anyone help?...
- 243
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3
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325
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Integral Representation of a Double Sum
Let us assume we know the value of $x$ and $y$. I'm trying to write the following double sum as an integral. I went through many pages and saw various methods but I'm completely lost with my problem.
$...
user977415
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1
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54
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Question about this power series
I know that $\displaystyle\sum_{n=0}^{+\infty} \dfrac{(-1)^nx^{2n+1}}{2n+1}$ is the power series expansion for $f(x) = \tan^{-1}x$. The interval of convergence for this series is $(-1,1]$.
If I ...
- 1,830
4
votes
2
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237
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General formula for the power sum $\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \mathbb{N}$
In my last question, it turns out to be solving the formula of $\sum_{k=1}^{n}k\omega^k$. I am curious if there is a geranal formula for the power sum:
$$\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \...
- 836
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I don't know how to solve this summation of series
Here is the question
$\sum_{x=1}^{\infty} 2^{x(t-1)}$, where $t$ is a constant
Compare this summation with $Z^c$, where $Z \geq 0$, and specify for what values of $Z$ and $c$, such that $\sum_{x=1}^{\...
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