CPM Homework Help : CALC3RD Problem 10-154

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10-154.

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

$\displaystyle\sum _ { k = 2 } ^ { \infty } ( - 1 ) ^ { k } \frac { k - 1 } { k ^ { 2 } - k }$
Step 1 (a):
$\displaystyle\sum_{k=1}^{\infty}\Big|(-1)^k\frac{k-1}{k^2-k}\Big|=\sum_{k=1}^{\infty}\frac{k-1}{k(k-1)}$
Step 2 (a):
$=\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}\text{ which diverges}$
Therefore series does not converge absolutely.
Does this series converge conditionally?

$\displaystyle\sum _ { k = 2 } ^ { \infty } \frac { ( - 1 ) ^ { k - 1 } } { \sqrt { k } - 1 }$
Step (b):
$\lim\limits_{k\to\infty}\frac{1}{\sqrt{k}-1}=0$
Therefore this series converges conditionally by the Alternating Series Test.
Does this series converge absolutely?

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( k + 1 ) ! } { \operatorname { ln } ( k + 1 ) }$
Step (c):
$\lim\limits_{k\to\infty}\frac{(k+1)!}{\ln(k+1)}=\infty$

$\displaystyle\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( k + 1 ) ! } { 2 ^ { 4 k } }$
Step (d):
$\lim\limits_{k\to\infty}\frac{(k+1)!}{2^{4k}}=?$

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