91Ó°ÊÓ

Evaluate the indefinite integral as an infinite series. \(\int \frac{e^{x}-1}{x} d x\)

\(47-51\) Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty} 4^{n} x^{n}$$

Evaluate the indefinite integral as an infinite series. \(\int \frac{\cos x-1}{x} d x\)

Evaluate the indefinite integral as an infinite series. \(\int \arctan \left(x^{2}\right) d x\)

\(47-51\) Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty} \frac{(x+3)^{n}}{2^{n}}$$

\(47-51\) Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty} \frac{(x+3)^{n}}{2^{n}}$$

Use series to approximate the definite integral to within the indicated accuracy. \(\int_{0}^{1} x \cos \left(x^{3}\right) d x \quad(\) three decimal places \()\)

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that $$\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)$$ is another series with this property.

Use series to approximate the definite integral to within the indicated accuracy. \(\int_{0}^{02}\left[\tan ^{-1}\left(x^{3}\right)+\sin \left(x^{3}\right)\right] d x\) (five decimal places)

\(53-54\) Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. $$\sum_{n=1}^{\infty} \frac{3 n^{2}+3 n+1}{\left(n^{2}+n\right)^{3}}$$