$(2)$

Figure $1$: Problem $5,$ Graphs of $y=1-x,y=3x-3x^2$

Final $1132,$ Fall $2024,$ December $9$th

Show that the function $f(x)=1-\sqrt[\phantom{2}]{1-x}$ has the following Taylor expansion around $x=0$ :

$f(x)={\displaystyle \underset{n=1}{\overset{}{}}}\frac{(\frac{2n}{n})x^n}{4^n(2n-1)}.$

Compute the sum of the series

${\displaystyle \underset{n=3}{\overset{}{}}}\frac{1}{n(n^2-4)}$

Use integration by parts to find reduction formulae for

$I_n={\displaystyle \underset{0}{\overset{}{}}}{e}^{-x}{x}^{n}sinxdx,J_n={\displaystyle \underset{0}{\overset{}{}}}{e}^{-x}{x}^{n}cosxdx.$

Use these reduction formulae to find $I_3$ and $J_3.$

$4.$ Use partial fractions to calculate ${\displaystyle \underset{1}{\overset{}{}}}\frac{x+1}{x^3+9x}dx.$

$5.$ Compute the volume of the solid generated by revolving the region bounded by $y=1-x$ and $y=3x-3x^2$ around the vertical line $x=\frac{1}{3}.$ Use only one method. For the other method just set up the integral but do not evaluate it$.$

$1$

$6.($a$)$ Show that the series ${\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n(n+1)}{n^2+1}$ is conditionally convergent.

$($a$)$ Use the comparison test idea to prove that

$($a$)$ Use the comparison test idea to prove that

${\displaystyle \underset{k=0}{\overset{}{}}}\frac{1}{{16}^k(4k+1)}{\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n(n+1)}{n^2+1}$

Find the value $of$

${\displaystyle \underset{k=0}{\overset{}{}}}\frac{1}{{16}^k(4k+1)}$