Write a partial sum for the power series,

[ani9890]

Write a partial sum for the power series, URGENT

Consider the function ln(1+4x).
Write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. For example, if the series were Sigma from n=0 to infinity of 3^nx^2n , you would write 1+3x2+3^2x^4+3^3x^6+3^4x^8. Also indicate the radius of convergence.

I got a power series = Sigma from n=0 to infinity of [(-1)^n(4^2n+1)(x^n+1)] / n+1
I got partial sum = 4x-32x^2+(1024/3)x^3-(16384/4)x^4+(262144/5)x^5
and radius of convergence = 1/4

radius of convergence is correct. But it says I have the partial sum wrong?
please help!

 

[Dickfore]

Consider the function ln(1+4x).
I got a power series = Sigma from n=0 to infinity of [(-1)^n(4^2n+1)(x^n+1)] / n+1

This is wrong!

Consider the simpler function:
$$\ln(1 + y) = \sum_{n = 0}^{\infty}{\frac{(-1)^{n} y^{n + 1}}{n + 1}}$$
Then, take $y = 4 x$. What does:
$$(4 \, x)^{n + 1} = ?$$
equal to?