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10-113.

Determine if the following series converge. State the test you used.

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } - 1 }$
Hint (a):
How does this series compare to the following series?
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{2^n}= \displaystyle\sum_{n=1}^{\infty}\Big(\frac{1}{2}\Big)^n$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ( n + 1 ) } { ( n + 2 ) ( n + 3 ) }$
Partial Answer (b):
This series diverges. Why?

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } + n } { n ! + 2 }$
Step (c):
$\lim\limits_{n\to\infty}\frac{2^{n+1}+n+1}{(n+1)!+2}\cdot\frac{n!+2}{2^n+n}$

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