$In$ this problem, you will derive the Taylor series expansion for $ln(1+x)$ using the

geometric series

$\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{}{}}}{x}^n,|x|<mn>1</mn>$

$(a)$ Find the Taylor series for $\frac{1}{1+x}by$ manipulating the geometric series.

$(b)$ Integrate your result from part $(a)$ term$-by-$term $to$ find the Taylor series for

$x,$ valid for $ln(1+x){\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{n}-1.$

$(c)$ Determine the radius $of$ converge for the Taylor series $ofln(1+x).$ Does the Taylor

Series converge $at$ the end$-$points?

$(d)$ Using your series from part $(b),$ find the exact value $of$ the alternating harmonic

series:

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{n}$