Test the series for convergence.

$$\sum_{n=1}^{\infty} e^{-n}$$

Write down the first four partial sums for the series.

$$\sum_{i=1}^{\infty}\left(\frac{2}{3}\right)^i$$

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

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

Find the general solution of the differential equations. $3 y^{\prime \prime}-4 y^{\prime}+y=0$.

Show that the series converges, using the comparison test for series with positive terms.

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

For which $x$ do the series converge?

$$\sum_{n=1}^{\infty} \frac{3}{n^2} x^n$$

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.

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