91Ó°ÊÓ

How large must \(N\) be in order for \(S_{N}=\sum_{k=1}^{N}(1 / k)\) just to exceed 4? Note: Computer calculations show that for \(S_{N}\) to exceed \(20, \quad N=272,400,600\), and for \(S_{N}\) to exceed 100 , \(N \approx 1.5 \times 10^{43}\)

Show that $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left[\frac{1}{1+(k / n)^{2}}\right] \frac{1}{n}=\frac{\pi}{4} $$

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \(\exp \left(x^{2}\right)\)

Determine the order \(n\) of the Maclaurin polynomial for \(4 \tan ^{-1} x\) that is required to approximate \(\pi=4 \tan ^{-1} 1\) to five decimal places, that is, so that \(\left|R_{n}(1)\right| \leq 0.000005\).

Show that the graph of \(y=x \sin \frac{\pi}{x}\) on \((0,1]\) has infinite length.

Using the definition of limit, prove that \(\lim _{n \rightarrow \infty} n /(n+1)\) \(=1 ;\) that is, for a given \(\varepsilon>0\), find \(N\) such that \(n \geq N \Rightarrow|n /(n+1)-1|<\varepsilon\).

Find the third-order Maclaurin polynomial for \((1+x)^{1 / 2}\) and bound the error \(R_{3}(x)\) for \(-0.5 \leq x \leq 0.5\).

Give conditions on \(p\) that determine the convergence or divergence of \(\sum_{n=1}^{\infty} \frac{1}{n^{p}}\left(1+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\cdots+\frac{1}{n^{p}}\right)\).

Prove that if \(\Sigma a_{n}\) diverges and \(\Sigma b_{n}\) converges, then \(\Sigma\left(a_{n}+b_{n}\right)\) diverges.

Find the third-order Maclaurin polynomial for \((1+x)^{3 / 2}\) and bound the error \(R_{3}(x)\) if \(-0.1 \leq x \leq 0\).