f) The series ∑_{j=1}^{∞}(–1)^{j}
sin j / j^{3 }converges
by the comparison test, because |(–1)^{j}
sin j / j^{3}|
≤
1/ j^{3 }and
the
series ∑_{j=1}^{∞
}1/
j^{p }converges
for any p > 1. Here p = 3.

g)
The series ∑_{n=2}^{∞
}1/n(ln
n)^{2 }converges
since the integral ∫_{2}^{∞
}1/x(ln
x)^{2 }dx
= – 1/ln x ]_{2}^{∞
}= 1/ln 2 converges
(note that lim_{x→+∞}1/ln
x = 0).

Apr 27th, 2015

I noticed that you have also asked about your third question (e) and got a wrong answer. Here is a solution.

Apply the ratio test to the series
∑_{n=0}^{∞}
[n6^{n+1 }/ n!5^{n }].