2. Let z be a real number with a 1.

(a) Prove the identity

2¹ =

-2-2

(Start from the complicated RHS and simplify to get the LHS!)

1

(b) Use (a) and the method of differences to prove the formula

Σ

for the sum of the first (n+1) terms of the geometric progression

1,2,²,...,"

(c) In calculus (which I'm not assuming you've seen!) you make sense

of infinite sums. For example, the recurring decimal 0.5 means

Σ(Η)

Using the formula from (b) plus the rigorous definition of infinite

sum from calculus, it follows that

9 9 9

0.9+0.09+0.009+=; + +

10 100 1000

Σ

i=0

providing |z| < 1. Assuming this formula, explain why it implies

that

0.5 1.