91Ó°ÊÓ

Prove Theorem D as follows: Let $$ f(x)=1+\sum_{n=1}^{\infty}\left(\begin{array}{l} p \\ n \end{array}\right) x^{n} $$ (a) Show that the series converges for \(|x|<1\). (b) Show that \((1+x) f^{\prime}(x)=p f(x)\) and \(f(0)=1\). (c) Solve this differential equation to get \(f(x)=(1+x)^{p}\).

Prove that if \(a_{n} \geq 0, b_{n}>0, \lim _{n \rightarrow \infty} a_{n} / b_{n}=\infty\), and \(\Sigma b_{n}\) diverges then \(\sum a_{n}\) diverges.

Show that a conditionally convergent series can be rearranged so as to diverge.

Suppose that \(\lim _{n \rightarrow \infty} n a_{n}=1 .\) Prove that \(\sum a_{n}\) diverges.

Find \(\lim _{n \rightarrow \infty} u_{n}\) of Problem 37 algebraically. Hint: Let \(u=\lim _{n \rightarrow \infty} u_{n} .\) Then, since \(u_{n+1}=\sqrt{3+u_{n}}, u=\sqrt{3+u}\). Now square both sides and solve for \(u\).

Show that \(\lim _{n \rightarrow \infty} a_{n}=0\) is not sufficient to guarantee the convergence of the alternating series \(\Sigma(-1)^{n+1} a_{n} .\) Hint: Alternate the terms of \(\sum 1 / n\) and \(\Sigma\left(-1 / n^{2}\right)\).

Suppose that Mary rolls a fair die until a "6" occurs. Let \(X\) denote the random variable that is the number of tosses needed for this " 6 " to occur. Find the probability distribution for \(X\) and verify that all the probabilities sum to 1 .

Let $$ f(t)=\left\\{\begin{array}{ll} 0 & \text { if } t<0 \\ t^{4} & \text { if } t \geq 0 \end{array}\right. $$ Explain why \(f(t)\) cannot be represented by a Maclaurin series. Also show that, if \(g(t)\) gives the distance traveled by a car that is stationary for \(t<0\) and moving ahead for \(t \geq 0, g(t)\) cannot be represented by a Maclaurin series.

Discuss the convergence or divergence of $$ \begin{array}{r} \frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1}+ \\\ \frac{1}{\sqrt{4}-1}-\frac{1}{\sqrt{4}+1}+\cdots \end{array} $$

Let $$ f(x)=\left\\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ (a) Show that \(f^{\prime}(0)=0\) by using the definition of the derivative. (b) Show that \(f^{\prime \prime}(0)=0\). (c) Assuming the known fact that \(f^{(n)}(0)=0\) for all \(n\), find the Maclaurin series for \(f(x)\). (d) Does the Maclaurin series represent \(f(x) ?\) (e) When \(a=0\), the formula in Theorem \(\mathrm{B}\) is called Maclaurin's Formula. What is the remainder in Maclaurin's Formula for \(f(x)\) ? This shows that a Maclaurin series may exist and yet not represent the given function (the remainder does not tend to 0 as \(n \rightarrow \infty)\)