91Ó°ÊÓ

Suppose that \(a_{n} > 0\) and \(b_{n} > 0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\Sigma a_{n}\) converges, can anything be said about \(\Sigma b_{n} ?\) Give reasons for your answer.

Suppose that \(a_{n} > 0\) and \(\lim _{n \rightarrow \infty} n^{2} a_{n}=0 .\) Prove that \(\sum a_{n}\) converges.

Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}\left(\frac{e}{\pi}\right)^{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\tanh n$$

Make up an infinite series of nonzero terms whose sum is a. 1 b. \(-3 \quad\) c. 0

Make up a geometric series \(\sum a r^{n-1}\) that converges to the number 5 if a. \(a=2\) b. \(a=13 / 2\)

Sequences generated by Newton's method \(\quad\) Newton's method, applied to a differentiable function \(f(x),\) begins with a starting value \(x_{0}\) and constructs from it a sequence of numbers \(\left\\{x_{n}\right\\}\) that under favorable circumstances converges to a zero of \(f .\) The recursion formula for the sequence is $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$ a. Show that the recursion formula for \(f(x)=x^{2}-a, a>0\) can be written as \(x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2\). b. Starting with \(x_{0}=1\) and \(a=3,\) calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.