91Ó°ÊÓ

Find the first four terms of the Taylor series for the functions. \((1+x)^{1 / 3}\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n^{2}}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$x^{2} \cos \left(x^{2}\right)$$

Find a formula for the \(n\)th term of the sequence. \(\frac{1}{9}, \frac{2}{12}, \frac{2^{2}}{15}, \frac{2^{3}}{18}, \frac{2^{4}}{21}, \ldots\)

Which of the series in Exercises converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} \ln n}$$

The estimate \(\sqrt{1+x}=1+(x / 2)\) is used when \(x\) is small. Estimate the error when \(|x|<0.01\).

Find the series' radius of convergence. $$\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^{2}} x^{n}$$

Use the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\) to obtain the Maclaurin series for \(\sin ^{2} x .\) Then differentiate this series to obtain the Maclaurin series for \(2 \sin x \cos x .\) Check that this is the series for \(\sin 2 x\).

a. Use Taylor's formula with \(n=2\) to find the quadratic approximation of \(f(x)=(1+x)^{k}\) at \(x=0\) ( \(k\) a constant). b. If \(k=3,\) for approximately what values of \(x\) in the interval [0,1] will the error in the quadratic approximation be less than \(1 / 100 ?\)

Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{n}$$