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An infinite series is the sum of an infinite sequence of numbers. When you come across the expression \(\sum_{k=1}^{\infty} a_k\), it represents adding up all terms \(a_k\) from an infinite sequence starting from \(k=1\) and going on forever. Infinite series are fascinating because, despite having infinitely many terms, they can often add up to a finite value.

This happens due to each term becoming smaller and smaller, eventually approaching zero. It is important to identify whether a series converges (adds up to a finite number) or diverges (adds to infinity or doesn't settle on a finite number). In the given exercise, you worked with geometric series, which are a type of infinite series where each subsequent term is obtained by multiplying the previous term by a constant number called the 'common ratio'.

Geometric series have a very useful property: they converge only when the absolute value of the common ratio is less than 1.

The term 'common ratio' refers to a constant value by which each term in a geometric sequence is multiplied to get the next term. It is a fundamental characteristic of a geometric series. In the series \((\frac{1}{6})^k\), the common ratio \(r_1\) is \(\frac{1}{6}\) which means each term is one-sixth of the previous term.

Identifying the common ratio is crucial because it indicates whether the series can be summed to a finite value. For a geometric series to converge, the common ratio must satisfy the condition \(|r| < 1\). This ensures that the terms in the series get smaller quickly enough to sum up to a finite number. In the second series from the exercise, \((\frac{1}{3})^{k-1}\), the common ratio \(r_2\) is \(\frac{1}{3}\). Both series fulfill the convergence criterion, allowing us to calculate their finite sums using a specific formula.

The series sum formula provides a straightforward way to find the sum of an infinite geometric series. For a series that starts with a first term \(a\) and has a common ratio \(r\), the sum \(S\) of the infinite series is given by:\[S = \frac{a}{1-r}, \quad \text{provided} \; \; |r| < 1\]
This formula arises because the series converges as the terms grow smaller due to the shrinking effect of repeatedly multiplying by \(r\).

In this exercise, you applied this formula twice: once for each geometric series. With the calculated sums \(S_1 = \frac{1}{5}\) and \(S_2 = \frac{3}{2}\), you simply added these to find the total sum of the given series. This illustrates how powerful the series sum formula is for evaluating infinite series with convergent geometric terms.