SOLUTION: Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite “sum to infinity” exists if and only if r < 1 and show that in this case the sum to infin
Algebra ->
Sequences-and-series
-> SOLUTION: Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite “sum to infinity” exists if and only if r < 1 and show that in this case the sum to infin
Log On
Question 549782: Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite “sum to infinity” exists if and only if r < 1 and show that in this case the sum to infinity is . Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website! We prove that a convergent sum exists by evaluating the limit as the number of terms approaches infinity:
Since r^(n+1) tends to zero, the sum converges to 1/(1-r). However, this only holds when |r| < 1. Otherwise, the limit diverges.
