21–42. Geometric series Evaluate each geometric series or state that it diverges.  


25.∑ (k = 0 to ∞) 0.9ᵏ

Verified step by step guidance

1. Identify the type of series given. The series is a geometric series of the form \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
2. Determine the first term \(a\) and the common ratio \(r\). Here, \(a = 0.9^0 = 1\) and \(r = 0.9\).
3. Check the convergence condition for a geometric series. A geometric series converges if and only if \(|r| < 1\).
4. Since \(|0.9| < 1\), the series converges. Use the formula for the sum of an infinite geometric series: \(S = \frac{a}{1 - r}\).
5. Substitute the values of \(a\) and \(r\) into the formula to express the sum: \(S = \frac{1}{1 - 0.9}\). This expression represents the sum of the series.

Key Concepts

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It is expressed as ∑ ar^k, where a is the first term and r is the common ratio.

Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges.

Sum Formula for Infinite Geometric Series

For a convergent infinite geometric series with first term a and common ratio r (|r| < 1), the sum is given by S = a / (1 - r). This formula allows quick evaluation of the series without summing infinitely many terms.