Part a$)$ Evaluate the improper integral. Enter an exact answer $($i$.$e$.$ no decimals$).$ Enter $$ if the series diverges to infinity. Enter $-$ if the series diverges to negative infinity. Enter DNE if the series diverges for some other reason.

${\displaystyle \underset{1}{\overset{}{}}}\frac{1}{x^3}dx=$

$0$

Part b$)$ What does the value of the improper integral tell us about the convergence of ${\displaystyle \underset{i=1}{\overset{}{}}}\frac{1}{i^3}?$

the series diverges

the series converges

the integral test does not apply

Part c$)$ If we use $2$ terms to approximate ${\displaystyle \underset{i=1}{\overset{}{}}}\frac{1}{i^3},$ what is the maximum error in our approximation? Give an exact answer $($i$.$e$.$ no decimals$).$

Error $$

Part c$)$ If we use the partial sum ${\displaystyle \underset{i=1}{\overset{n}{}}}\frac{1}{i^3}$ to approximate ${\displaystyle \underset{i=1}{\overset{}{}}}\frac{1}{i^3},$ how large should we need choose $n$ to guarantee accuracy to within $0\mathrm{.}001?$

$n=$