The sum of an infinite geometric series with first term \( a \) and common ratio \( r \) (where \( |r| < 1 )\) is given by: \( \displaystyle S = \frac{a}{1 – r} \). In this post, we will learn how to prove it.
Proof:
Consider the infinite geometric series:
\( S = a + ar + ar^2 + ar^3 + \dotsb (1)\)
Multiply both sides by \( r \):
\( rS = ar + ar^2 + ar^3 + ar^4 + \dotsb (2)\)
Subtract the second equation from the first:
\( \begin{aligned} &S – rS = a \\ &\Rightarrow S(1 – r) = a \\ &\Rightarrow S = \frac{a}{1 – r} \end{aligned} \)Condition for Convergence:
The series converges if and only if \( |r| < 1 \). If \( |r| \geq 1 \), the sum diverges.