91Ó°ÊÓ

Show that if \(a_{n} \rightarrow 0\) and the sequince with terms \(b_{n}\) is bounded, then \(a_{n} b_{n} \rightarrow 0\)

Give the first six terms of the sequence and then give the \(n\) th term. $$a_{1}=1 ; \quad a_{n+1}=2 a_{n}+1$$.

Show that the hyperbolic arc \(y=(b / a) \sqrt{x^{2}-a^{2}}\) is asymptotic to the line \(y=(b / a) x\) as \(x \rightarrow \infty\)

Suppose that \(a_{n} \rightarrow L .\) Show that if \(a_{n} \leq M\) for all \(n\), then \(L \leq M\)

The sequence defined recursively by setting $$a_{\mathrm{n}+2}=a_{n+1}+a_{n} \quad \text { starting with } \quad a_{1}=a_{2}=1$$,is called the Fibonacci sequence. (a) Calculate \(a_{3}, a_{4}, \cdots, a_{10}\) (b) Define $$r_{n}=\frac{a_{n+1}}{a_{n}}$$,$$\text { Calculate } r_{1}, r_{2}, \cdots, r_{6}$$, (c) Assume that \(r_{n} \rightarrow L\), and find \(L\). HINT: Rclatc \(r\), to \(r_{n}, 1\).

Give the first six terms of the sequence and then give the \(n\) th term. $$a_{1}=1 ; \quad a_{n+1}=a_{n}+\cdots+a_{1}$$.

Use comparison test (11.7.2) to determine whether the integral converges. $$\int_{0}^{\infty}\left(1+x^{5}\right)^{-1 / 6} d x$$

Give the first six terms of the sequence and then give the \(n\) th term. $$a_{1}=3 ; \quad a_{n+1}=4-a_{n}$$.

Show that the hyperbolic arc \(y=(b / a) \sqrt{x^{2}-a^{2}}\) is asymptotic to the line \(y=(b / a) x\) as \(x \rightarrow \infty\)

The figure shows an angle \(\theta\) between 0 and \(\pi / 2 .\) Let \(T(\theta)\) be the area of triangle \(A B C\), and let \(S(\theta)\) be the arca of the segment of the circle cut by the chord \(A B\). Find \(\lim _{\theta \rightarrow \infty} T(\theta) / S(\theta)\)