2 let z be a real number with a 1 a prove the identity 2 2 2 start fro
2. Let z be a real number with a 1.
(a) Prove the identity
(Start from the complicated RHS and simplify to get the LHS!)
(b) Use (a) and the method of differences to prove the formula
for the sum of the first (n+1) terms of the geometric progression
(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
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
providing |z| < 1. Assuming this formula, explain why it implies
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