Define the quantities in the form $a+b i$.

$$e^{(3 \pi / 2) i}$$

Test the given series for convergence. If it can be summed using a geometric series, do so.

$$\sum_{i=1}^{\infty} \frac{3^{i+1}}{5^{i-1}}$$

Use the integral test to define the convergence or divergence of the series.

$$\sum_{i=2}^{\infty} \frac{1}{i(\ln i)^{3 / 2}}$$

Find solutions of the given equation in the form of power series: $y=\sum_{i=0}^{\infty} a_i x^i$. $y^{\prime \prime}+2 x y^{\prime}=0$.

Test for convergence.

$$\sum_{k=2}^{\infty} \frac{\cos k \pi}{\ln k}$$

Test the series for convergence.

$$1+\frac{1}{2}+\underbrace{\frac{1}{4}+\frac{1}{4}}_2+\underbrace{\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}}_4+\underbrace{\frac{1}{16}+\cdots}_8$$

Using the method of the earlier Example, find the quantities.

$$\sqrt{-16 i}$$

Write power series representations for the functions. $\tan ^{-1} x$ and its derivative. [Hint: Do the derivative first.]

Test the series for convergence.

$$\sum_{j=1}^{\infty} \frac{\sin j}{2^j}$$

Test for convergence.

$$\sum_{m=1}^{\infty} \frac{(-1)^m(m+1)}{m^2+1}$$

Write power series representations for the functions.

$$(d / d x) e^{-x^2}$$

Test the series for convergence.

$$\sum_{i=1}^{\infty} \frac{6}{7+7 i}$$

Using the method of the earlier Example, find the quantities.

$$\sqrt{9 i}$$

Find the general solution to the given equation ( $y$ is a function of $x$ or $x$ is a function of $t$ as appropriate). $y^{\prime \prime}-2 \sqrt{2} y^{\prime}+2 y=\cos x-e^{-x}$.