Test for convergence.

$$\sum_{n=1}^{\infty} \frac{\sqrt{n}+n+n^{3 / 2}}{\sqrt{n}+n+n^{5 / 2}+n^3}$$

Show that the series $\sum_{j=1}^{\infty}\left(1-2^{-j}\right) / j$ diverges.

Test for convergence.

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

Find the series for $1 /[(1-x)(2-x)]$ by writing

$$\frac{1}{(1-x)(2-x)}=\frac{A}{1-x}+\frac{B}{2-x}$$

and adding the resulting geometric series.

Test the series for convergence.

$$\sum_{i=2}^{\infty} \frac{1}{\ln i}$$

Test for convergence.

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

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

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

Test the series for convergence.

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

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

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

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.

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

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 $a+b i$.

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