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. $$0.3+0.03+0.003+\cdots$$

Short answer

Answer: We can conjecture that the value of the infinite series is \(\frac{1}{3}\).

Step-by-step solution

Identify the first term (a) and the common ratio (r)

The given infinite series is $$0.3+0.03+0.003+\cdots$$ The first term (a) is 0.3, and the common ratio (r) can be found by dividing a consecutive term by the previous one. $$\text{Common ratio } (r)= \frac{0.03}{0.3} = \frac{1}{10}$$

Find the first four terms of the sequence of partial sums (S1, S2, S3, S4)

To find the first four partial sums, we sum up the terms of the series up to the desired term. $$S_1 = 0.3$$ $$S_2 = 0.3 + 0.03$$ $$S_3 = 0.3 + 0.03 + 0.003$$ $$S_4 = 0.3 + 0.03 + 0.003 + 0.0003$$ Now, let's find the value of each of the partial sums. $$S_1 = 0.3$$ $$S_2 = 0.3 + 0.03 = 0.33$$ $$S_3 = 0.3 + 0.03 + 0.003 = 0.333$$ $$S_4 = 0.3 + 0.03 + 0.003 + 0.0003 = 0.3333$$

Conjecture about the value of the infinite series

After finding the first four terms of the sequence of partial sums, we can observe that the values are getting closer to \(\frac{1}{3}\). Since the common ratio is between -1 and 1, i.e., \(|r|<1\), the infinite sum converges to a limit. Using the formula for the sum of an infinite geometric series, we can find the value of the infinite series. $$S_{\infty} = \frac{a}{1 - r} = \frac{0.3}{1 - \frac{1}{10}} = \frac{0.3}{\frac{9}{10}} = \frac{1}{3}$$ So, the conjecture is that the value of the infinite series = \(\frac{1}{3}\).