I'm actually having real hard time with this problem.... I tried Ratio test on ii) but r became 1 which means you can't use it(or I solved it wrong...)....
Determine whether the series $\sum_{n=1}^\infty a_n$ an defined by the following formulas coverge or diverge.
i) $a_1$=$1\over2$, $a_{n+1}=(a_n)^{n+1}$
ii) $a_1$=$1\over2$, $a_{n+1} $=${n+\ln n}\over {n+10}$ $a_n$
Someone save me!
A few hints:
I. Notice that for $n\ge 1$, if $0\le a_n\le 1$, then $0\le(a_n)^{n+1}\le a_n$. It should be possible from there to show $a_n$ is positive and monotonic decreasing. The ratio test can be applied from there, since we know that $\frac{a_{n+1}}{a_n}=(a_n)^n$.
II. For sufficiently large $n$, the ratio $\frac{n+\ln(n)}{n+10}$ will be larger than one. This means that for large $n$, $a_{n+1}>a_n$ provided the terms are positive. Consider what this means for the series.