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


33.∑ (k = 4 to ∞) 1 / 5ᵏ

Verified step by step guidance

1. Identify the first term \( a \) of the geometric series by substituting the starting index \( k = 4 \) into the term \( \frac{1}{5^k} \). So, \( a = \frac{1}{5^4} \).
2. Determine the common ratio \( r \) of the geometric series. Since the terms are of the form \( \frac{1}{5^k} \), the ratio between consecutive terms is \( \frac{1}{5} \).
3. Check the convergence of the series by verifying if the absolute value of the common ratio \( |r| < 1 \). If this condition holds, the series converges; otherwise, it diverges.
4. If the series converges, use the formula for the sum of an infinite geometric series starting at \( k = 0 \): \[ S = \frac{a}{1 - r} \]. Since our series starts at \( k = 4 \), \( a \) is already the first term at \( k=4 \).
5. Substitute the values of \( a \) and \( r \) into the sum formula to express the sum of the series. This will give the sum in terms of powers of 5 without calculating the final numeric value.

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure helps in identifying and evaluating the series.

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 and does not have a finite sum.

Index Shift in Series Summation

When a series starts at an index other than zero or one, it may be necessary to adjust the formula for the sum by rewriting the series with a shifted index. This helps in correctly identifying the first term a and applying the geometric series sum formula.