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. $$4+0.9+0.09+0.009+\cdots$$

Short answer

Answer: The value of the infinite series is 5.

Step-by-step solution

Find the first term and common ratio

The first term of the series is given in the exercise: \(a_1 = 4\). To find the common ratio, we can divide the second term by the first term or the third term by the second term. Let's divide the second term by the first term: $$r = \frac{0.9}{4} = 0.225$$ The common ratio is 0.225.

Calculate the first four partial sums

The partial sums are found by adding the terms of the sequence up to a certain point. We will calculate the first four partial sums: 1. \(S_1 = a_1 = 4\) 2. \(S_2 = a_1 + a_2 = 4 + 0.9 = 4.9\) 3. \(S_3 = a_1 + a_2 + a_3 = 4.9 + 0.09 = 4.99\) 4. \(S_4 = a_1 + a_2 + a_3 + a_4 = 4.99 + 0.009 = 4.999\) The first four partial sums are 4, 4.9, 4.99, and 4.999.

Make a conjecture about the value of the infinite series

As we can see from the first four partial sums, the terms of the series are getting closer and closer to 5. We can make a conjecture that the value of the infinite series is 5. To support our conjecture, we can use the formula for the sum of an infinite geometric series: $$S_\infty = \frac{a_1}{1 - r}$$ We already found the first term \(a_1 = 4\) and the common ratio \(r = 0.225\). Let's plug these values into the formula: $$S_\infty = \frac{4}{1 - 0.225} = \frac{4}{0.775} = 5.16129$$ However, since the series 0.09, 0.009, … is actually an infinite geometric series with first term, \(a_1 = 0.9\) and common ratio, \(r = 0.1\), it also converges, so its sum is: $$S_\infty = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1$$ Therefore, the infinite series converges to sum of 4 added to the sum of the series 0.09, 0.009,… . So, the sum is: $$4+1 = 5$$ Our conjecture was correct, and the value of the infinite series is 5.