91Ó°ÊÓ

Let \(f(x)=(1+x)^{p}\), where \(p\) is a constant, \(p \neq 0,1,2,3,4,5\). (a) Compute the third degree Taylor polynomial for \(f(x)\) around \(x=0\). (b) Compute the fth degree Taylor polynomial for \(f(x)\) around \(x=0\).

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=3}^{\infty} \frac{k}{2 k^{3}-2}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\cos \left(x^{2}\right) $$

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}-n}}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=3^{x} $$

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=x^{2} \cos (-x) $$

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{n=2}^{\infty} \frac{n+1}{\ln n}\)

Determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=2}^{\infty} \frac{2^{k}}{5^{k}-5}\)

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=\cos ^{2} x $$

(a) Let \(\sum_{k=1}^{\infty} a_{k}\) be a convergent series with \(0