91Ó°ÊÓ

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x^{2} \sin (x / 2) $$

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x^{3}+\cos \left(x^{2}\right) $$

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty} 1 / n^{2} $$

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+\sin (n)} $$

Use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} $$

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If the partial sums of an infinite series diverge, then the series __ diverge.

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{2^{3 n}}{3^{2 n}}\)

Determine whether the given series converges absolutely, converges conditionally, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{1+1 / n}{2}\right)^{n} $$

In Exercises \(27-38\), compute the Taylor polynomial \(T_{N}\) of the given function \(f\) with the given base point \(c\) and given order \(N\). \(f(x)=\cos (x)\) \(N=4 \quad c=\pi / 3\)

Use the Root Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\)