Question

Find the values of x for which the series converges. (enter your answer using interval notation.) ∞ (x + 9)n n = 1 find the sum of the series for those values of x.

Answer 1

If the series is

$\displaystyle\sum_{n=1}^\infty(x+9)^n$


then it is geometric, which we know converges as long as the base is smaller than 1 in absolute value, i.e. $|x+9|$. "Expanding" the absolute value, this is equivalent to

$-1$

When $x$ is in this interval, the geometric sum converges to

$\displaystyle\sum_{n=1}^\infty(x+9)^n=\frac1{1-(x+9)}=-\frac1{x+8}$