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$4.(24$pts$)$ Find the radius of convergence and the interval of convergence for the following power series. You will get no credit for an answer that does not have legible work supporting it$.$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(x-3{)}^n}{2^n\sqrt[\phantom{2}]{n}}$

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$5.(10$pts$)$ Write the following as a power series and find its interval of convergence.

Power Series:

Interval of Convergence:

$\frac{2x}{8-x^3}$

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$6.(8$pts$)$ You know the Mclauren Series for $sinx.$ In this problem I would like you to demonstrate how we found it using the Mclauren Series formula. Make sure you show enough steps $($it is easy not to when you already know the answer!$)$

$7.(6$pts$)$ Write the first $6$ terms of the McLaurin Series $($using the formula, not from scratch$)$ for $sin{x}^2.$

$8.(8$pts$)$ Estimate the value of ${\displaystyle \underset{0}{\overset{1}{}}}sin(x^2)dx$ with an error You can't integrate this directly, you must use the series expansion you found in #$8!$

${\displaystyle \underset{0}{\overset{1}{}}}sin(x^2)dx$~~ $$ with error

Work supporting your answer: