91Ó°ÊÓ

To determine if a geometric series converges, it's crucial to examine the common ratio. The common ratio (\( r \)) plays a pivotal role in this evaluation. For a geometric series to converge, the absolute value of the common ratio must be less than one:

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

This condition for convergence ensures that as more terms are added, the successive terms get smaller and smaller, approaching zero. This shrinking effect causes the series to sum to a finite value. In the exercise provided, the common ratio is \( e^{-2} \). Since exponentials that are negative shrink, \( e^{-2} \) is definitely less than one:

• \(0 < e^{-2} < 1\)
• Thus, the series converges.

The common ratio (\( r \)) of a geometric series is the factor by which we multiply a term to obtain the next term. This value is crucial as it determines the behavior of the series:

• A positive value that is less than one indicates a converging series.
• A value more than one or negative indicates divergence in many contexts.

To find the common ratio, divide any term by its preceding term. In the sample problem, the series is given by \( \sum_{k=1}^{\infty} e^{-2k} \). Consequently:

• \( r = \frac{e^{-2(k+1)}}{e^{-2k}} = e^{-2} \).

This shows that each term is a constant fraction \( e^{-2} \) of the prior one. Recognizing the common ratio helps not only to confirm convergence but also aids in calculating the sum of the series when it converges.

When a geometric series converges, we can calculate its sum using a special formula. This formula needs two elements: the first term of the series (\( a_1 \)) and the common ratio (\( r \)). The sum of an infinite geometric series is given by:

• \( S_{\infty} = \frac{a_1}{1-r} \)

This formula derives from the fact that as the number of terms approaches infinity, the effect of each successive term decreases.In the exercise, the first term of the series is \( a_1 = e^{-2} \), and the common ratio remains \( e^{-2} \). Substituting these into the formula:

• \( S_{\infty} = \frac{e^{-2}}{1 - e^{-2}} \)

Thus, this formula efficiently calculates the sum of the series. Understanding how each component within this equation influences the series helps demystify the process of summing infinite geometric series.