Question

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

Answer 1

Answer:

Step-by-step 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

$\frac{a(r^n-1}{r-1},|r|>1\\\frac{1-r^2}{1-r},|r|$

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