Question

In 2-3 sentences describe why the only way an infinite geometric series has a sum is if |r| < 1.

Answer 1

Explanation:

A geometric series is one where each term is multiplied by a constant value known as r to get the next term

Sum of n terms of a geometric series is

`(a(r^n-1)/(r-1),|r|>1\\(1-r^2)/(1-r),|r|<1`

Sum of infinite series is obtained as the limiting value of this sum when n tends to infinity

We find that only when |r|<1, r power n tends to 0 as n tends to infinity.

Other r power n diverges.

Hence geometric series infinite sum formula is valid only when

|r|<1 since the series sum converges to a finite value