I have to find the series expansion and interval of convergence for the function $\ln(1 – x)$.
For the expansion, I have gone through the process and obtained the series:
$-x – (x^2/2) – (x^3/3) – . . . – (-1)^k((-x)^k)/k$
I know that the interval of convergence will be $(-1,1)$, but am having trouble with the ratio test component to achieve this result. i.e. I am having trouble breaking down/simplifying the equation.
Thanks very much
Best Answer
Don't let the $(-1)^k$ or $(-x)^k = (-1)^kx^k$ trouble you. They have the effect of canceling each other out for odd $k$, and besides, for the ratio test, we apply it taking the absolute value of the general term $|a_k|$.
$$|a_k| = \frac{(x)^k }{k}$$
$$\frac{a_{k+1}}{a_k} = \frac{\frac{(x)^{k+1}}{k+1}}{\frac{(x)^k }{k}} = \frac{xk}{k+1}$$