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


35.∑ (k = 0 to ∞) 3(–π)^(–k)

Verified step by step guidance

1. Identify the first term \( a \) of the geometric series by substituting \( k = 0 \) into the general term \( 3(-\pi)^{-k} \). This gives \( a = 3(-\pi)^0 = 3 \).
2. Determine the common ratio \( r \) by finding the factor that each term is multiplied by to get the next term. This is \( r = \frac{3(-\pi)^{-1}}{3(-\pi)^0} = (-\pi)^{-1} = \frac{1}{-\pi} = -\frac{1}{\pi} \).
3. Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| -\frac{1}{\pi} \right| = \frac{1}{\pi} \). Since \( \pi > 1 \), \( |r| < 1 \), so the series converges.
4. Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to express the sum, where \( a = 3 \) and \( r = -\frac{1}{\pi} \).
5. Write the sum explicitly as \( S = \frac{3}{1 - \left(-\frac{1}{\pi}\right)} = \frac{3}{1 + \frac{1}{\pi}} \). Simplify this expression to get the sum in a more compact form.

Key Concepts

Geometric Series Definition

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio, r. It has the form ∑ ar^k, where a is the first term and k is the index. Understanding this structure is essential to identify and evaluate the series.

Convergence Criteria for Infinite Geometric Series

An infinite geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion determines whether the sum approaches a finite value or not.

Sum Formula for Convergent Geometric Series

When a geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a quick way to find the total sum of infinitely many terms.