#### Answer

$\sum_{k=1}^{100}(-1)^{k+1}* k* x^{k-1}$

#### Work Step by Step

We write the sum in sigma notation:
(We notice that the terms have the form $(integer)x^{integer-1}$ with consecutive integers and alternating signs.)
$1-2x+3x^{2}-4x^{3}+5x^{4}+ \cdots-100x^{99}=\sum_{k=1}^{100}(-1)^{k+1}* k* x^{k-1}$