Test the series for convergence.

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

Find the imaginary part of the complex numbers.

$$\frac{1+i}{i}$$

Test for convergence.

$$\sum_{j=1}^{\infty} \frac{(j+1)^{100}}{j !}$$

Test the series for convergence.

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

Test for convergence.

$$\sum_{j=1}^{\infty}(-1)^j \sin \left(\frac{\pi}{4 j}\right)$$

Test the series for convergence. $1+\underbrace{\frac{1}{4}+\frac{1}{4}}_2+\underbrace{\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}}_4+\underbrace{\frac{1}{64}+\frac{1}{64}+\cdots}_8$.

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

$$\sqrt{\sqrt{i}}$$

Write power series representations for the functions. The second derivative of $1 /(1-x)$.

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

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

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

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

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

Write down the first four partial sums for the series.

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

For which $x$ do the series converge?

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

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

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