Problem 104 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

Step-by-Step Solution

Answer

It makes sense to use a formula to find the sum of this infinite geometric series, and the sum is 4.5. However, it does not make sense to check the answer by directly adding all the terms of an infinite series, as this would be impossible.

Step 1: Identify the structure of series

Notice the given series \(3 + 1 + \frac{1}{3} + \frac{1}{9} + \cdots \) is a geometric series where each successive term is \(\frac{1}{3}\) times the preceding term. Here, the initial term \(a = 3\) and the common ratio \(r = \frac{1}{3}\).

Step 2: Apply the infinite geometric series formula

The formula for the sum of an infinite geometric series \(S\) when \(|r| < 1\) is given by \(S = \frac{a}{1-r}\). Substituting the known values \(a = 3\) and \(r = \frac{1}{3}\) into the formula gives \(S = \frac{3}{1 - \frac{1}{3}} = \frac{3}{\frac{2}{3}} = 4.5\)

Step 3: Comment on direct addition of terms

The statement about checking the answer by actually adding all terms does not make sense, as it is impossible to add 'all' terms of an infinite series. An infinite series, by its very nature, has an infinite number of terms; we can't physically add all of them up. Therefore, the claim of validating the answer using such a method is not logical.