Question

In Exercises 85 and \(86,\) (a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1+x+x^{2}+x^{3}+\cdots $$

Short answer

The common ratio of the geometric series is \(x\), the function that gives the sum of the series is \(S = 1 / (1 - x)\), and graphing this function along with the partial sums \(S_3\) and \(S_5\) shows that as more terms are added to the series, the partial sums get closer to the value of the function \(S = 1 / (1 - x)\).

Step-by-step solution

Find the Common Ratio

To find the common ratio of the geometric series, divide one term by the previous term. Looking at the given series \(1 + x + x^2 + x^3 + \cdots\) it can be seen that each term is multiplied by \(x\) to get the next term. Therefore, the common ratio \(r\) is \(x\).

Write the Function For the Sum

The sum \(S\) of an infinite geometric series is given by the formula \(S = a / (1 - r)\), where \(a\) is the first term and \(r\) is the common ratio. For the given series, \(a = 1\) and \(r = x\), so the sum is given by the function \(S = 1 / (1 - x)\).

Graph the Function and Partial Sums

This step involves using a graphing utility to graph the sum function found in step 2, as well as the partial sums \(S_3\) and \(S_5\). The partial sum \(S_3 = 1 + x + x^2\), and \(S_5 = 1 + x + x^2 + x^3 + x^4\). By plotting these values on the graph, one can compare how the series converges to the sum as more terms are added.