How does Sn = 1 + t + t^2 + ... + t^(n-1) = (1-t^n)/(1-t)?

Question

The original question is "show that if t∈R with modulus of t < 1, then the series ∑_n=1^inf t^n-1= 1+t+t²+... coverges.

Answer 1

We have 

 

   S_n       = 1 + t + t² + ··· + t^n-1 

 

tS_n      =        t + t² + ··· + t^n-1 + tⁿ 

———————————————

(1 - t)S_n = 1                              - tⁿ 

 

So (1 - t)S_n = 1 - tⁿ, which implies

 

S_n = (1 - tⁿ)/(1 - t).