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


21.∑ (k = 0 to ∞) (1/4)ᵏ

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 \(r\) from the series \(\sum_{k=0}^{\infty} \left(\frac{1}{4}\right)^k\). Here, \(a = 1\) (since when \(k=0\), the term is \(\left(\frac{1}{4}\right)^0 = 1\)) and \(r = \frac{1}{4}\).
3. Check the convergence of the series by examining the absolute value of the common ratio \(r\). The geometric series converges if and only if \(|r| < 1\). Since \(|\frac{1}{4}| = \frac{1}{4} < 1\), the series converges.
4. Use the formula for the sum of an infinite geometric series that converges: \(S = \frac{a}{1 - r}\). Substitute \(a = 1\) and \(r = \frac{1}{4}\) into this formula.
5. Write the expression for the sum of the series as \(S = \frac{1}{1 - \frac{1}{4}}\). This expression represents the sum of the infinite geometric 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 direct computation of the series sum without adding infinitely many terms.