1) Find the values of the following series (the last two by partial fractions).

$$a) \sum_2^\infty \frac{1}{n-1} \qquad \qquad b) \sum_2^\infty \frac{1}{n(n-1)} \qquad \qquad \hfill b) \sum_1^\infty \frac{4n+2}{n^2(n+1)^2}$$

2) Determine whether the following series are convergent or divergent. Show your reasoning.

i)

$$\sum_1^\infty \frac{1}{\sqrt{n(n+1)}}$$

ii)

$$\sum_1^\infty \frac{1}{n(\ln n)^s} \eqno (s$$    some number$$)$$

iii)

$$\sum_1^\infty n e^{-n^2}$$

iv)

$$\sum_1^\infty \frac{1+\sqrt{n}}{(n+1)^3-1}$$

v)

$$\sum_1^\infty (n^{1/n} -1)^n$$

vi)

$$\sum_1^\infty \frac{(n!)^2}{2^{n^2}}$$

vii)

$$\sum_1^\infty e^{-n^2}$$

viii)

$$\sum_1^\infty \frac{n^{n+1/n}}{(n+1/n)^n} \eqno ($$Tricky!$$)$$

ix)

$$\sum_2^\infty \frac{1}{(\ln n)^{1/n}}$$

x)

$$\sum_1^\infty \frac{2^n n!}{n^n}$$

xi)

$$\sum_1^\infty \frac{3^n n!}{n^n}$$