8)
i)
ii)
9)
i) The series
∞∑n=14nxn=∞∑n=1(4x)n
converges when |4x|<1. That is, if -1/4 < x < 1/4 the given geometric series converges. To find the sum of the series, we shift the index:
∞∑n=1(4x)n=∞∑n=0(4x)n+1=(4x)∞∑n=0(4x)n=4x1-4x
The other two can be done in the same way.
