Problem 6 - Calculus Early Transcendentals

Question

What is the condition for convergence of the geometric series \(\sum_{k=0}^{\infty} a r^{k} ?\)

Step-by-Step Solution

Answer

Answer: The condition for convergence of a geometric series is the absolute value of the common ratio, \(r\), must be less than 1, i.e., \(|r|<1\).

Step 1: Recall the formula for the sum of an infinite geometric series

The sum of an infinite geometric series is given by the formula: \(\sum_{k=0}^{\infty} a r^{k} = \frac{a}{1-r}\), but only if the series converges.

Step 2: Determine the condition under which the infinite geometric series converges

For the series to converge, the denominator \((1-r)\) in the formula should never be zero. Moreover, we need the terms in the series to become smaller and smaller as \(k\) increases, so that the sum approaches a finite value. This can only happen if \(|r| < 1\). In other words, the absolute value of the common ratio, \(r\), must be less than 1.

Step 3: State the condition for convergence

The condition for convergence of the geometric series \(\sum_{k=0}^{\infty} a r^{k}\) is: \(|r|<1\).