  ### Home > APCALC > Chapter 10 > Lesson 10.4.1 > Problem10-154

10-154.

Determine if each of the following series converges absolutely, converges conditionally, or diverges. State the tests you used.

1. $\displaystyle\sum _ { k = 2 } ^ { \infty } ( - 1 ) ^ { k } \frac { k - 1 } { k ^ { 2 } - k }$

$\displaystyle\sum_{k=1}^{\infty}\Big|(-1)^k\frac{k-1}{k^2-k}\Big|=\sum_{k=1}^{\infty}\frac{k-1}{k(k-1)}$

$=\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}\text{ which diverges}$

Therefore series does not converge absolutely.

Does this series converge conditionally?

1. $\displaystyle\sum _ { k = 2 } ^ { \infty } \frac { ( - 1 ) ^ { k - 1 } } { \sqrt { k } - 1 }$

$\lim\limits_{k\to\infty}\frac{1}{\sqrt{k}-1}=0$

Therefore this series converges conditionally by the Alternating Series Test.

Does this series converge absolutely?

1. $\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( k + 1 ) ! } { \operatorname { ln } ( k + 1 ) }$

$\lim\limits_{k\to\infty}\frac{(k+1)!}{\ln(k+1)}=\infty$

1. $\displaystyle\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( k + 1 ) ! } { 2 ^ { 4 k } }$

$\lim\limits_{k\to\infty}\frac{(k+1)!}{2^{4k}}=?$