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Chapter 9: Problem 71

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Short Answer

Expert verified
Answer: The limit of the sequence of partial sums is 1.

Step by step solution

01

Identify the common ratio and first term of the geometric series

In the given infinite series, $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$, we can identify the first term and common ratio: - The first term (when \(k=1\)) is \(a_1 = 9(0.1)^1 = 0.9\) - The common ratio is \(r = 0.1\)
02

Write out the first four terms of the sequence of partial sums

We can calculate the first four partial sums (denoted as \(S_1, S_2, S_3, S_4\)) using the formula for the sum of a geometric series: $$S_n = a_1 \frac{1 - r^n}{1-r}$$ Using the identified first term and common ratio, let's calculate the first four partial sums: 1. \(S_1 = \frac{0.9(1-0.1^1)}{0.9} = 0.9\) 2. \(S_2 = \frac{0.9(1-0.1^2)}{0.9} \approx 0.99\) 3. \(S_3 = \frac{0.9(1-0.1^3)}{0.9} \approx 0.999\) 4. \(S_4 = \frac{0.9(1-0.1^4)}{0.9} \approx 0.9999\)
03

Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist

Since the given series is a geometric series with \(|r|<1\), it converges. To find the limit of the sequence \({S_n}\), we can use the formula for the sum of an infinite geometric series: $$S_\infty = \lim_{n\to\infty} S_n = \frac{a_1}{1-r}$$ Plugging in the values for \(a_1\) and \(r\): $$S_\infty = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1$$ So, the limit of the sequence of partial sums \({S_n}\) is 1.

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Most popular questions from this chapter

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Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k}$$
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