Question

# Find the value of x for which the series converges \sum_{n=1}^\infty(x+2)^n Find the sum of the series for those values of x.

Series
Find the value of x for which the series converges
$$\sum_{n=1}^\infty(x+2)^n$$ Find the sum of the series for those values of x.

2021-06-03

Consider the series $$\sum_{n=1}^\infty(x+2)^n$$
Let $$a_n=(x+2)^n,\ a_{n+1}=(x+2)^{n+1}$$
$$\frac{a_{n+1}}{a_n}=\frac{(x+2)^{n+1}}{(x+2)^n}$$
$$=x+2$$
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}(|x+2|)=|x+2|$$
Using ratio test, this will converge, when
$$|x+2|<1$$
$$-1<(x+2)<1$$
$$-3$$
So, the series is converges at the values lies in the interval, (-3,-1)
And $$\sum_{n=1}^\infty(x+2)^n=(x+2)^1+(x+2)^2+(x+2)^3+...+(x+2)^\infty$$
$$=\frac{x+2}{1-(x+2)}$$
$$=\frac{x+2}{-1-x}$$