What is the sum of the geometric series \( S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \) if \( r=1 \) ? Make a conjecture with regard to \( r^{n} \) and \( S_{n} \) as \( n \rightarrow \infty \)

For a geometric series with general term a·r^(k) (starting at k = 0), the sum of the first n terms is   Sₙ = a + a·r + a·r² + … + a·r^(n–1). When r = 1, every term in the series equals a (since 1 raised to any power is 1). Thus, the sum becomes   Sₙ = a + a + a + … + a  (n times)      = n · a. Now let’s examine what happens as n → ∞. 1. For the term rⁿ: When r = 1, we have 1ⁿ = 1 for any n. Therefore, as n gets arbitrarily large, rⁿ remains 1. 2. For the sum Sₙ: Since Sₙ = n·a, as n → ∞, if a is not 0, the sum increases without bound. In other words, unless a is 0, Sₙ diverges to ∞ (or –∞ if a is negative). The only exception is when a = 0, in which case every term is zero and the sum remains zero. Conjecture:  • If r = 1, then rⁿ always equals 1 regardless of n.  • Moreover, Sₙ = n·a. Thus, as n → ∞, unless a = 0, the sum Sₙ → ∞ (or diverges in the sense that it does not converge to a finite limit). This behavior contrasts with the case when |r| < 1, where rⁿ → 0 and Sₙ converges to a finite value (namely, Sₙ → a/(1 – r) as n → ∞).