The function is represented as a power series
Find the first few coefficients in the power series.
I've found c0 to be 6 in this case, and the radius of convergence to be 0.1. Can someone help me find c1, c2, c3, and c4? Please help!
The function is represented as a power series
Find the first few coefficients in the power series.
I've found c0 to be 6 in this case, and the radius of convergence to be 0.1. Can someone help me find c1, c2, c3, and c4? Please help!
$\displaystyle \frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-1)^{n}x^{2n}$
So,
$\displaystyle \frac{1}{1+(10x)^2}=\sum_{n=0}^{\infty}(-1)^{n}(10x)^{2n}$
$\displaystyle \frac{6}{1+(10x)^2}=\sum_{n=0}^{\infty}(-1)^{n}6(10x)^{2n}$
$\displaystyle \text{But }\sum_{n=0}^{\infty}(-1)^{n}6(10x)^{2n} = \sum_{n=0}^{n=\infty} c_n x^n$
$\displaystyle \text{Thus }\sum_{n=0}^{\infty}6(-100)^{n}(x)^{2n} = \sum_{n=0}^{n=\infty} c_n x^n$
So clearly for an even n $\displaystyle c_n = 6(-100)^{\frac{n}2} $ and for an odd n $\displaystyle c_n = 0 $