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


39.∑ (k = 2 to ∞) (–0.15)ᵏ

Verified step by step guidance

1. Identify the first term of the geometric series by substituting the lower limit of the summation into the general term: the first term is \(a = (-0.15)^2\).
2. Determine the common ratio \(r\) of the geometric series, which is the base of the exponent in the term, so \(r = -0.15\).
3. Check the convergence of the series by evaluating the absolute value of the common ratio: if \(|r| < 1\), the series converges; otherwise, it diverges.
4. Since the series starts at \(k=2\) instead of \(k=0\) or \(k=1\), express the sum from \(k=2\) to infinity in terms of the sum from \(k=0\) to infinity by factoring out the first two terms or adjusting the formula accordingly.
5. Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term of the series starting at \(k=2\), to write the sum expression without calculating the final numerical value.

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. Understanding the structure helps determine convergence and sum.

Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion is essential to decide whether the series has a finite sum or not.

Sum Formula for Convergent Geometric Series

When an infinite geometric series converges, its sum is given by S = a / (1 - r), where a is the first term and r is the common ratio. This formula allows direct calculation of the series sum once convergence is established.