Question

Imvestigation In Exercise 7 you found that the interval of convergence of the geometric series \(\sum_{n=0}^{\infty}\left(\frac{x}{2}\right)^{n}\) is (-2,2) . (a) Find the sum of the series when \(x=\frac{3}{4}\). Use a graphing utility to graph the first six terms of the sequence of partial sums and the horizontal line representing the sum of the series. (b) Repeat part (a) for \(x=-\frac{3}{4}\). (c) Write a short paragraph comparing the rate of convergence of the partial sums with the sum of the series in parts (a) and (b). How do the plots of the partial sums differ as they converge toward the sum of the series? (d) Given any positive real number \(M,\) there exists a positive integer \(N\) such that the partial sum \(\sum_{n=0}^{N}\left(\frac{3}{2}\right)^{n}>M\) Use a graphing utility to complete the table. $$ \begin{array}{|l|l|l|l|l|} \hline \boldsymbol{M} & 10 & 100 & 1000 & 10,000 \\ \hline \boldsymbol{N} & & & & \\ \hline \end{array} $$

Short answer

The sum of the series when \(x=\frac{3}{4}\) and \(x=-\frac{3}{4}\) is \(\frac{8}{5}\) and \(-\frac{8}{5}\) respectively. For M = 10, N can be 5. For M = 100, N can be 7. For M = 1000, N can be 9. And for M = 10,000, N can be 11.

Step-by-step solution

Calculate the sum of the series

We will use the formula for the sum of an infinite geometric series, which is \(\frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio. Here, for \(x=\frac{3}{4}\), the first term is 1 and the common ratio is \(r=\frac{3}{4}/2=\frac{3}{8}\). Hence, the sum is \(\frac{1}{1-\frac{3}{8}}=\frac{8}{5}\) for \(x=\frac{3}{4}\), and similarly \(\frac{8}{5}\) for \(x=-\frac{3}{4}\).

Graph the first six terms of the sequence of partial sums

You should graph the first six partial sums for \(x = \frac{3}{4}\) and \(x = -\frac{3}{4}\). It would show that for \(x = \frac{3}{4}\), the series is converging toward the sum \(\frac{8}{5}\) and for \(x = -\frac{3}{4}\), it is converging to \(-\frac{8}{5}\). An additional element that should be added to these graphs is a horizontal line representing these sums.

Compare the rate of convergence of the partial sums

The rate of convergence appears to be faster for \(x=-\frac{3}{4}\) than for \(x=\frac{3}{4}\), as the negative values make the sum fluctuate around the final value in the initial terms than the sequence with positive values.

Fill in the table

We need to find integer values of \(N\) such that \( \sum_{n=0}^{N}\left(\frac{3}{2}\right)^{n}>M \). For M = 10, N can be 5. For M = 100, N can be 7. For M = 1000, N can be 9. And for M = 10,000, N can be 11.