91Ó°ÊÓ

Use the power series expansion for \(\left(e^{x}-1\right) / x\) to verify that $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1 $$

Express the repeating decimal as a fraction. $$ 0.72727272 \ldots $$

Let \(a \neq 0\), and assume that \(\lim _{n \rightarrow \infty} a_{n}=a\) and \(a_{n} \neq 0\) for all \(n\). Show that \(\sum_{n=1}^{\infty}\left|a_{n+1}-a_{n}\right|\) converges if and only if \(\sum_{n=1}^{\infty}\left|\frac{1}{a_{n+1}}-\frac{1}{a_{n}}\right|\) converges.

Find the smallest positive integer \(n\) for which \(\left|a_{n}\right|<\varepsilon\). $$ a_{n}=1-\sqrt[n]{1.2} ; \varepsilon=10^{-3} $$

Express the repeating decimal as a fraction. $$ 0.024242424 \ldots $$

Find a power series expansion for \(\left(e^{x}-1-x\right) / x^{2}\) and use it to evaluate $$ \lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}} $$

Find the smallest positive integer \(n\) for which \(\left|a_{n}\right|<\varepsilon\). $$ a_{n}=1-\sqrt[n]{n} ; \varepsilon=2 \times 10^{-2} $$

In Theorem \(9.4\) we have \(\lim _{n \rightarrow \infty} a_{n}=\lim _{x \rightarrow \infty} f(x)=\) \(\lim _{x \rightarrow 0^{+}} f(1 / x) .\) In Exercises 43-44 find an appropriate function \(f\) corresponding to the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\), and plot \(f(1 / x)\). Then guess \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=n \sinh e^{-n} $$

Express the repeating decimal as a fraction. $$ 0.232232232 \ldots $$

Express \(\ln [(1+x) /(1-x)]\) as a power series.