91Ó°ÊÓ

A geometric series is a series where each term after the first is found by multiplying the previous term by a constant, known as the common ratio, denoted by \( r \). This type of series can either converge or diverge depending on the value of \( r \).
For a geometric series \( \sum_{k=0}^{\infty} ar^k \) to converge, the absolute value of the common ratio must be less than one, \(|r| < 1\). When it converges, the series has a finite sum that can be calculated using the formula:

• \( S = \frac{a}{1-r} \) where \( a \) is the first term of the series, and \( r \) is the common ratio.

In the provided exercise, we had two geometric series:

• The first with \( r = \frac{1}{6} \) and a first term \( a = \left(\frac{1}{6}\right)^1 \).
• The second with \( r = \frac{1}{3} \) and a first term \( a = 1 \) since \( \left(\frac{1}{3}\right)^{0} = 1 \).

Both series have \( |r| < 1 \), and thus both converge.

Convergence refers to the idea that as more terms are added to an infinite series, the sum approaches a specific limit. For geometric series, convergence is determined by the common ratio \( r \).
If \( |r| < 1 \), the infinite series will converge to a finite value. This is because the terms get smaller and smaller, eventually making an insignificant contribution to the sum. Conversely, if \( |r| \geq 1 \), the series will diverge, meaning it does not approach a finite limit.
In our exercise, both sub-series have common ratios \( \frac{1}{6} \) and \( \frac{1}{3} \), both of which are less than 1 in absolute value, ensuring their convergence. Thus, each sub-series reaches their individual finite sums which we could then add together to find the convergence value of the original series given as \( \frac{8}{5} \).

Series evaluation involves summing up the terms of a series. When the series converges, this often means finding the sum to which the series approaches as more and more terms are included.
For geometric series, we utilize the formula \( S = \frac{a}{1-r} \) to determine the sum if convergence criteria are met.
In our solution:

• The first series sum was calculated as \( \frac{1}{5} \) using \( r = \frac{1}{6} \).
• The second series sum was calculated as \( \frac{3}{2} \) using \( r = \frac{1}{3} \).
• By adding these sums, we find the total series evaluation: \( \frac{1}{5} + \frac{3}{2} = \frac{8}{5} \).

Thus, after evaluating both geometric series, we concluded that their combined sum, \( \frac{8}{5} \), represents the series's overall value when converged.