91Ó°ÊÓ

Find the Maclaurin series for \(f(x),\) and state the radius of convergence. $$ f(x)=\ln (2+x) $$

The famous Fibonacci sequence is defined recursively by \(a_{k+1}=a_{k}+a_{k-1}\) with \(a_{1}=a_{2}=1\) (a) Find the first ten terms of the sequence. (b) The terms of the sequence \(r_{k}=a_{k+1} / a_{k}\) give approximations to \(\tau,\) the golden ratio. Approximate the first ten terms of this sequence. (c) Assuming that \(\lim _{n \rightarrow x} r_{n}=\tau,\) prove tha $$ \tau=\frac{1}{2}(1+\sqrt{5}) $$

Exer. 49-50: For the given convergent series, (a) approximate \(S_{1}, S_{2},\) and \(S_{3}\) to five decimal places and (b) approximate the sum of the series to three decimal places. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}}{e^{\left(n^{2}\right)}} $$

Suppose \(\sum a_{n}\) and \(\sum b_{n}\) are positive-term series. Prove that if \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=0\) and \(\sum b_{n}\) converges, then \(\sum a_{n}\) converges. (This is not necessarily true for series that contain negative terms.)

Find the Maclaurin series for \(f(x),\) and state the radius of convergence. $$ f(x)=(1+x)^{2 / 3} $$

Find the Maclaurin series for \(f(x),\) and state the radius of convergence. $$ f(x)=\frac{1}{\sqrt{1-x^{2}}} $$

Prove that if \(\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=x\) and \(\sum b_{n}\) diverges, then \(\sum a_{n}\) diverges.

Show that \(\cos x \approx 1-\frac{1}{2} x^{2}+\frac{1}{24} x^{4}-\frac{1}{720} x^{6}\) is accurate to five decimal places if \(0 \leq x \leq \pi / 4\).

Find a series representation for \(e^{-x}\) in powers of \(x+2\).

Prove or disprove: If \(\sum a_{n}\) and \(\sum b_{n}\) both diverge, then \(\sum\left(a_{n}+b_{n}\right)\) diverges.