Question

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

Based on the given infinite geometric series with the general term \(a_k = 9(0.1)^k\), we found the first four terms of the sequence to be \(0.9, 0.09, 0.009,\) and \(0.0009\). The first four terms of the sequence of partial sums are \(0.9, 0.99, 0.999,\) and \(0.9999\). Since the common ratio, \(r = 0.1\), is less than 1, we can conclude that the sum converges. Using the formula for the sum of an infinite geometric series, we estimated the limit of the sequence of partial sums to be \(1\).

Step-by-step solution

Identify the given information

We are given the general term of the infinite geometric series: \(a_k = 9(0.1)^k\). Notice that our general term has the form \(a_k = Ar^k\) where \(A = 9\) and \(r = 0.1\). Now, we can identify the first four terms of the sequence: \ \(a_1 = 9(0.1)^1 = 0.9\), \ \(a_2 = 9(0.1)^2 = 0.09\), \ \(a_3 = 9(0.1)^3 = 0.009\), \ \(a_4 = 9(0.1)^4 = 0.0009\).

Find the first four partial sums

Recall that the \(n\)-th partial sum of a series is the sum of the first \(n\) terms of the series. We will find the first four partial sums using the terms we found in step 1. \(S_1 = a_1 = 0.9\), \ \(S_2 = a_1 + a_2 = 0.9 + 0.09 = 0.99\), \ \(S_3 = a_1 + a_2 + a_3 = 0.9 + 0.09 + 0.009 = 0.999\), \ \(S_4 = a_1 + a_2 + a_3 + a_4 = 0.9 + 0.09 + 0.009 + 0.0009 = 0.9999\). Thus, the first four terms of the sequence of partial sums are \(0.9, 0.99, 0.999,\) and \(0.9999\).

Estimate the limit

We need to find the limit of the sequence of partial sums, \(S_n\). We know that for an infinite geometric series, the sum converges if the common ratio \(r\) has an absolute value less than 1. In our case, the common ratio \(r = 0.1\), which is less than 1. So, the sum converges. To estimate the limit, we will use the formula for the sum of an infinite geometric series: \ $$S = \frac{A}{1-r},$$ \ where \(A\) is the first term of the series and \(r\) is the common ratio. Substitute \(A = 9(0.1) = 0.9\) and \(r = 0.1\) into the formula: \ $$S = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1.$$ Therefore, the limit of the sequence of partial sums is \(1\).

Key concepts

Partial Sums

Understanding partial sums is essential in grasping the concept of an infinite geometric series. A partial sum, denoted as S_n, is the sum of the first n terms of a sequence. Imagine it like slowly adding slices until you eventually get the whole pie. In the case of our exercise, we calculated the first four partial sums of the series by adding the terms computed from the general term, a_k = 9(0.1)^k.

The sequence of the first four partial sums was found to be S₁ = 0.9, S₂ = 0.99, S₃ = 0.999, and S₄ = 0.9999. Notice how with each additional term, the sum gets closer to the next decimal place of 1, hinting at the behavior of the series as we add more and more terms. This process helps us see the pattern within the series and begins to reveal the nature of its convergence.

Sequence Convergence

When discussing sequence convergence in the context of geometric series, we're basically asking: does this series settle down to a single value, or does it bounce around indefinitely? The exact point where a series settles is called its limit. In our exercise, we noted that the sequence of partial sums gradually approaches 1, suggesting that the series does indeed converge.

To confirm convergence, we check the common ratio, r. The rule of thumb is simple: if |r| < 1, the series converges. Since our common ratio is 0.1, well within this bound, we can confidently say the series converges. This means that as we sum up an infinite number of terms, the total doesn't grow without bounds but approaches a finite limit.

Limit of a Series

So, what is the magical number that our series is inching toward? This is where we discuss the limit of a series. The limit is, in a way, the final destination for the sequence of partial sums if we were to continue adding terms forever. For an infinite geometric series, there's a straightforward formula to find this limit: S = A / (1 - r), where A is the first term and r is the common ratio.

In our exercise, after substituting the values A = 0.9 and r = 0.1 into the formula, we find that the limit is exactly 1. The elegance of this discovery lies in the method: we didn't have to sum an infinite number of terms manually. Instead, the formula gave us a shortcut to determine the limit efficiently, confirming that the infinite sum of the series is precisely 1.