https://img705.imageshack.us/img705/...at24032013.jpg

8)

i)
https://img51.imageshack.us/img51/1254/mat24032013.png

ii)
https://img801.imageshack.us/img801/...at24032013.png

9)

i) The series

∞

∑

n=1

4ⁿxⁿ=

∞

∑

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=

4x

1-4x

The other two can be done in the same way.