Question

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the answer is arn interval, enter your answer using interval notation. If the answer is a finite set of values, enter your answers as a comma separated list of values.)

[infinity]

Σ (-1)n+1(x-8)/n8n

n=1

Answer 1

I'm guessing the power series given is

$\displaystyle \sum_{n=1}^\infty (-1)^{n+1}\frac{(x-8)^n}{8^n}$

which you can condense somewhat into

$\displaystyle -\sum_{n=1}^\infty \left(\frac{8-x}8\right)^n$

You should recognize this as a geometric series, which converges for

$\left|\dfrac{8-x}8\right|<1 \implies |8-x|<8 \implies \boxed{0 < x < 16}$