Question

For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. $$0.6+0.06+0.006+\cdots$$

Short answer

Answer: The value of the infinite geometric series is \(\frac{2}{3}\).

Step-by-step solution

Identify the common ratio and the first term

We are given an infinite geometric series: $$0.6+0.06+0.006+\cdots$$ The first term, denoted by \(a\), is 0.6. The common ratio, denoted by \(r\), can be found by dividing the second term by the first term: $$r = \frac{0.06}{0.6} = \frac{1}{10}$$.

Find the first four partial sums

To find the first four terms of the sequence of partial sums, we can simply compute the sums: 1. \(S_1 = 0.6\) 2. \(S_2 = 0.6 + 0.06 = 0.66\) 3. \(S_3 = 0.66 + 0.006 = 0.666\) 4. \(S_4 = 0.666 + 0.0006 = 0.6666\) So, the first four partial sums are 0.6, 0.66, 0.666, and 0.6666.

Make a conjecture about the value of the infinite series

To make a conjecture about the value of the infinite series, let's first check if the series converges by examining the common ratio \(r\). Since \(|r| = |\frac{1}{10}|<1\), the series converges. Now we can use the formula for the sum of an infinite geometric series, which is: $$S_\infty = \frac{a}{1 - r}$$ Substitute the given values for \(a\) and \(r\): $$S_\infty = \frac{0.6}{1 - \frac{1}{10}} = \frac{0.6}{\frac{9}{10}}$$ Finally, compute the sum: $$S_\infty = \frac{0.6 * 10}{9} = \frac{6}{9} = \frac{2}{3}$$ Based on the calculations, it can be conjectured that the value of the infinite series is \(\frac{2}{3}\).