91Ó°ÊÓ

Find a geometric power series for the function, centered at 0 , (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{1+x} $$

Explain why the Integral Test does not apply to the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} $$

In Exercises \(17-20\), approximate the sum of the series by using the first six terms. (See Example 4.) $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} 3}{n^{2}} $$

Use Theorem 7.11 to determine the convergence or divergence of the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{3}{n^{5 / 3}} $$

Find the Maclaurin series for the function. (Use the table of power series for elementary functions.) $$ f(x)=e^{x}+e^{-x}=2 \cosh x $$

In Exercises \(31-46,\) find the sum of the convergent series. $$ \sum_{n=2}^{\infty} \frac{1}{n^{2}-1} $$

In Exercises \(35-38,\) write an equivalent series with the index of summation beginning at \(n=1\). $$ \sum_{n=0}^{\infty} \frac{x^{n}}{n !} $$

In Exercises \(35-38,\) use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. $$ \cos (0.3) \approx 1-\frac{(0.3)^{2}}{2 !}+\frac{(0.3)^{4}}{4 !} $$

In Exercises \(35-38,\) write an equivalent series with the index of summation beginning at \(n=1\). $$ \sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !} $$

In Exercises \(35-38,\) use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. $$ \arctan (0.4) \approx 0.4-\frac{(0.4)^{3}}{3} $$