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


23.∑ (k = 0 to ∞) (–9/10)ᵏ

Verified step by step guidance

1. Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, the series is \( \sum_{k=0}^{\infty} \left(-\frac{9}{10}\right)^k \), so \( a = 1 \) (when \( k=0 \)) and \( r = -\frac{9}{10} \).
2. Recall the formula for the sum of an infinite geometric series: if \( |r| < 1 \), then the series converges and its sum is \( S = \frac{a}{1 - r} \). Otherwise, the series diverges.
3. Check the absolute value of the common ratio: \( |r| = \left| -\frac{9}{10} \right| = \frac{9}{10} < 1 \). Since this is true, the series converges.
4. Apply the sum formula by substituting \( a = 1 \) and \( r = -\frac{9}{10} \) into \( S = \frac{a}{1 - r} \). This gives \( S = \frac{1}{1 - \left(-\frac{9}{10}\right)} \).
5. Simplify the denominator to find the expression for the sum: \( 1 - \left(-\frac{9}{10}\right) = 1 + \frac{9}{10} \). This completes the setup to calculate the sum.

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 and Divergence of Infinite Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1, meaning the sum approaches a finite limit. If |r| ≥ 1, the series diverges and does not have a finite sum.

Sum Formula for Convergent Geometric Series

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