91Ó°ÊÓ

A certain ball has the property that each time it falls from a height \(h\) onto a hard, level surface, it rebounds to a height \(r h,\) where \(0

Show that the sequence defined by $$a_{1}=2 \quad a_{n+1}=\frac{1}{3-a_{n}}$$ satisfies \(0 < a_{n} \leqslant 2\) and is decreasing. Deduce that the sequence is convergent and find its limit.

Give an example of a pair of series \(\Sigma a_{n}\) and \(\Sigma b_{n}\) with posi- tive terms where \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=0\) and \(\Sigma b_{n}\) diverges, but \(\sum a_{n}\) converges. [Compare with Exercise \(42 . ]\)

For which positive integers \(k\) is the following series convergent? $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(k n) !}$$

Find the value of $$c\( if \)\sum_{n=2}^{\infty}(1+c)^{-n}=2$$

(a) Show that \(\sum_{n=0}^{\infty} x^{n} / n !\) converges for all \(x\). (b) Deduce that \(\lim _{n \rightarrow \infty} x^{n} / n !=0\) for all \(x\).

Evaluate the indefinite integral as an infinite series. $$\int \frac{\cos x-1}{x} d x$$

Find all positive values of \(b\) for which the series \(\sum_{n=1}^{\infty} b^{\ln n}\) converges.

Evaluate the indefinite integral as an infinite series. $$\int \arctan \left(x^{2}\right) d x$$

(a) Let \(a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldots\) \(a_{n+1}=f\left(a_{n}\right),\) where \(f\) is a continuous function. If \(\lim _{n \rightarrow \infty} a_{n}=L,\) show that \(f(L)=L\) (b) Illustrate part (a) by taking \(f(x)=\cos x, a=1,\) and estimating the value of \(L\) to five decimal places.