All Questions
5
questions
0
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1
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76
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Prove the formula $1+r\cdot \cos(α)+r^{2}\cos(2α)+\cdots+r^{n}\cos(nα)=\dfrac{r^{n+2}\cos(nα)-r^{n+1}\cos[(n+1)α]-r\cosα+1}{r^{2}-2r\cdot \cos(α)+1}$
For $r,a\in\mathbb{R}:\; r^{2}-2r\cos{a}+1\neq 0$ prove the formula $$1+r\cdot \cos(a)+r^{2}\cos(2a)+\cdots+r^{n}\cos(na)=\dfrac{r^{n+2}\cos(na)-r^{n+1}\cos[(n+1)a]-r\cdot \cos(a)+1}{r^{2}-2r\cdot \...
- 21
0
votes
1
answer
57
views
Find the radius of convergence of $\sum_{i=0}^\infty a_n$ where $\sum_{i=0}^\infty 2^n a_n$ converges, but $\sum_{i=0}^\infty (-1)^n2_na_n$ diverges [closed]
From this example: $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty|a_n|$ diverges. Then Radius of convergence?
I believe I'm supposed to leverage these two statements to show that $R \leq |z|$...
- 70
3
votes
2
answers
143
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Evaluate infinite s, series, similar to $\cos(z)$
Evaluate the sum
$$\sum_{k=0}^\infty\frac{(-1)^kz^{ak}}{\Gamma(1+ak)}$$
where $a\in \mathbb{R}_{>0}$ and $z\in\mathbb{C}$.
I know if $a=2$ then this is the series expansion for $\cos(z)$. But for ...
- 2,369
4
votes
6
answers
955
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Prove this formula $\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$
I am trying to use prove, by just simple algebraic manipulation, to prove the equality of this formula.
$$\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}= 1+\sum_{n=1}^{+\infty}r^{n}\cos\left(nx\right)$$
I have ...
- 3,187
4
votes
0
answers
824
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Laurent expansion for $\sqrt{z(z-1)}$
Let $f(z) = \sqrt{z(z-1)}$. The branch cut is the real interval $[0,1]$, and $f(z)>0$ for real $z$ that are greater than 1. I need to find the first few terms of the Laurent expansion of $f(z)$ for ...
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