  ### Home > APCALC > Chapter 10 > Lesson 10.4.2 > Problem10-168

10-168.

Determine the radius and interval of convergence for each of the following power series.

1. $\displaystyle \sum _ { k = 1 } ^ { \infty } \frac { ( - 4 ) ^ { k } ( x - 3 ) ^ { k } } { k }$

$\lim \limits_{k\to\infty}\Big|\frac{(-4)^{k+1}(x-3)^{k+1}}{k+1}\cdot\frac{k}{(-4)^k(x-3)^k}\Big|<1$

$|x-3|\lim \limits_{k\to\infty}\Big|\frac{4k}{k+1}\Big|<1$

$2\frac{3}{4}

Now test the convergence of the series at each endpoint of the interval.

1. $\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { 2 ^ { k } ( x + 4 ) ^ { k } } { ( k + 1 ) ! }$

$\frac{(k+1)!}{(k+2)!}=\frac{1}{k+2}$

1. $1 + 3 x + \frac { 9 x ^ { 2 } } { 2 } + \frac { 27 x ^ { 3 } } { 3 } + \frac { 81 x ^ { 4 } } { 4 } +\ldots$

$=\displaystyle \sum_{k=0}^\infty \frac{(3x)^k}{k}$

1. $\displaystyle\sum _ { k = 2 } ^ { \infty } \frac { x ^ { k } } { 3 ^ { k \operatorname { ln } ( k ) } }$

After applying the Ratio Test, you should get:

$3|x|<1$

Now test the convergence of the series using the endpoints of the interval.