91Ó°ÊÓ

In Exercises 1–4, 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. $$ f(x)=\frac{1}{2+x} $$

Verify the formula. \(\frac{(2 k-2) !}{(2 k) !}=\frac{1}{(2 k)(2 k-1)}\)

Finding the Center of a Power Series In Exercises \(1-4\) , state where the power series is centered. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} 1 \cdot 3 \cdot \cdot \cdot(2 n-1)}{2^{n} n !} x^{n} $$

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\left(-\frac{2}{5}\right)^{n} $$

Verify the formula. \(1 \cdot 3 \cdot 5 \cdot \cdot(2 k-1)=\frac{(2 k) !}{2^{k} k !}\)

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\sin \frac{n \pi}{2} $$

Finding the Center of a Power Series In Exercises \(1-4\) , state where the power series is centered. $$ \sum_{n=1}^{\infty} \frac{(x-2)^{n}}{n^{3}} $$

Using the Integral Test In Exercises \(1-22,\) confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}} $$

In Exercises 1–4, 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. $$ f(x)=\frac{4}{3+x} $$

Using the Direct Comparison Test In Exercises \(3-12\) , use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{2 n-1} $$