Find the values of x for which the series converges. (enter your answer using interval notation.)

Asked by maham237 @ 22/12/2020 in Mathematics viewed by 356 People

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.

Answered by maham237 @ 22/12/2020

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

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

Find the values of x for which the series converges. (enter your answer using interval notation.)

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