91Ó°ÊÓ

Use a graphing utility to show that \(\frac{\sqrt{8}}{9801} \sum_{n=0}^{\infty} \frac{(4 n) !(1103+26,390 n)}{(n !) 396^{4 n}}=\frac{1}{\pi}\) (Note: This series was discovered by the Indian mathematician Srinivasa Ramanujan in \(1914 .\) )

Let \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) (a) Find the interval of convergence of \(f\). (b) Show that \(f^{\prime}(x)=f(x)\). (c) Show that \(f(0)=1\). (d) Identify the function \(f\).

Use a computer algebra system to find the fifth-degree Taylor polynomial (centered at \(c\) ) for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. \(f(x)=\sin \frac{x}{2} \ln (1+x), \quad c=0\)

In your own words, state the difference between absolute and conditional convergence of an alternating series.

Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\Sigma b_{n}\) converges, \(\Sigma a_{n}\) also converges.

In general, how does the accuracy of a Taylor polynomial change as the degree of the polynomial is increased? Explain your reasoning.

show that the function represented by the power series is a solution of the differential equation. $$ y=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !}, \quad y^{\prime \prime}+y=0 $$

Use a computer algebra system to find the fifth-degree Taylor polynomial (centered at \(c\) ) for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. \(g(x)=\sqrt{x} \ln x, \quad c=1\)

Find the sum of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{3^{n}(2 n+1)} $$

Determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty} \frac{1}{4^{n}}\)