91Ó°ÊÓ

Use the binomial series to find the power series representation of the function. Then find the radius of convergence of the series. \(f(x)=\frac{x}{(1+x)^{2}}\)

If \(a\) is a constant, find the radius and the interval of convergence of the power series \(\sum_{n=0}^{\infty} a^{n}(x-c)^{n}\).

Find an approximation of the sum of the series accurate to two decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \cdot 2^{n}} $$

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(3 n) !}\)

Find the values of \(p\) for which the series is convergent. $$ \sum_{n=1}^{\infty} \frac{\ln n}{n^{p}} $$

If the radius of convergence of the power series \(\sum a_{n} x^{n}\) is \(R\) what is the radius of convergence of the power series \(\sum a_{n} x^{2 n}\) ?

Show that the series $$ \frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{9}+\frac{1}{8}-\frac{1}{27}+\cdots+\frac{1}{2^{n}}-\frac{1}{3^{n}}+\cdots $$ converges, and find its sum. Why isn't the Alternating Series Test applicable?

Determine whether the series is convergent or divergent. \(\sum_{n=0}^{\infty} \frac{1}{n !}\)

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\sqrt{n+1}-\sqrt{n}\)

Find all values of \(x\) for which the series \(\sum_{n=1}^{\infty} \frac{x^{n}}{n}\) (a) converges absolutely and (b) converges conditionally.