91Ó°ÊÓ

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

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. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}} $$

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

Finding the Center of a Power Series In Exercises \(1-4\) , state where the power series is centered. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} $$

Finding Partial Sums In Exercises \(1-6,\) find the sequence of partial sums \(S_{1}, S_{2}, S_{3}, S_{4},\) and \(S_{5}\) . $$ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\cdots $$

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. $$ \sum_{n=1}^{\infty} 3^{-n} $$

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\sin x, \quad c=\frac{\pi}{4} $$

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=\frac{3 n}{n+4} $$

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. $$ f(x)=\frac{2}{5-x} $$

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