Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each of the following geometric sums, rst write the sum using summation notation and then write the sum in closed form. (a) \(\frac{2}{3^{2}}+\frac{2}{3^{4}}+\frac{2}{3^{6}}+\cdots \frac{2}{3^{18}}\) (b) \(1-2+2^{2}-2^{3}+2^{4}-\cdots+2^{46}\) (c) \(-\frac{1}{100}+\frac{1.1}{100}-\frac{1.21}{100}+\frac{1.331}{100}-\cdots-\frac{1.1^{100}}{100}\) (d) \(\frac{2}{3^{2}}+\frac{2^{2}}{3^{3}}+\frac{2^{3}}{34}+\cdots+\frac{2^{16}}{317}\)

Step-by-Step Solution

Answer

(a): \(\Sigma_{n=2}^{18} \frac{2}{3^{2n}} \approx 0.02\) (b): \(\Sigma_{n=0}^{23} (-1)^n2^{2n} \approx -0.5\) (c): \(\Sigma_{n=1}^{100} (-1)^n\frac{1.1^n}{100}\) (this cannot be simplified) (d): \(\Sigma_{n=2}^{17} \frac{2^n}{3^{n+1}} \approx 0.4\)

Step 1 Solution for (a)

We need to express \(\frac{2}{3^{2}}+\frac{2}{3^{4}}+\frac{2}{3^{6}}+\cdots \frac{2}{3^{18}}\) using summation notation. It can be rewritten as \(\Sigma_{n=2}^{18} \frac{2}{3^{2n}}\). 'a' is \(\frac{2}{3^{2}}\), 'r' is \(\frac{1}{9}\), 'm' is 18 and 'k' is 2. Now, the closed form expression of the series can be obtained using the formula and is \(\frac{2}{3^{2}} \frac{1 - (\frac{1}{9})^{18-2+1}}{1 - \frac{1}{9}}\)

Step 2 Solution for (b)

Express \(1-2+2^{2}-2^{3}+2^{4}-\cdots+2^{46}\) using summation notation. This can be written as \(\Sigma_{n=0}^{23} (-1)^n2^{2n}\). 'a' is 1, 'r' is -4, 'm' is 23 and 'k' is 0. The closed form of the series is \(1\frac{1 - (-4)^{23+1}}{1 - (-4)}\)

Step 3 Solution for (c)

The series \(-\frac{1}{100}+\frac{1.1}{100}-\frac{1.21}{100}+\frac{1.331}{100}-\cdots-\frac{1.1^{100}}{100}\) can be written as \(\Sigma_{n=1}^{100} (-1)^n\frac{1.1^n}{100}\). 'a' is -\( \frac{1}{100}\), 'r' is -1.1, 'm' is 100 and 'k' is 1. This yields the closed form as \(-\frac{1}{100}\frac{1 - (-1.1)^{100+1}}{1 - (-1.1)}\)

Step 4 Solution for (d)

Express \(\frac{2}{3^{2}}+\frac{2^{2}}{3^{3}}+\frac{2^{3}}{34}+\cdots+\frac{2^{16}}{317}\) using summation notation. This can be written as \(\Sigma_{n=2}^{17} \frac{2^n}{3^{n+1}}\). 'a' is \( \frac{2}{9}\), 'r' is \( \frac{2}{3}\), 'm' is 17 and 'k' is 2. The closed form of the series is \(\frac{2}{9}\frac{1 - (\frac{2}{3})^{17-2+1}}{1 - \frac{2}{3}}\)

Key Concepts

Summation Notation

Summation notation is a concise way of representing the addition of a sequence of numbers. It is particularly useful for dealing with series, which are sums of terms that follow a specific pattern. In a geometric series, each term after the first is found by multiplying the previous term by a constant called the ratio. The symbol \( \Sigma \) (the Greek uppercase letter sigma) is used to represent summation. The general form is \( \Sigma_{i=m}^{n} a_{i} \) where \( a_{i} \) is the i-th term, \( m \) is the lower bound or the starting index, and \( n \) is the upper bound or ending index. For instance, in example (a) from our exercise, the summation notation for \( \frac{2}{3^{2}}+\frac{2}{3^{4}}+\dots+\frac{2}{3^{18}} \) is written as \( \Sigma_{n=2}^{18} \frac{2}{3^{2n}} \). The index of summation, \( n \) in this case, successively takes on each integer value from the lower bound to the upper bound, and the expression \( \frac{2}{3^{2n}} \) generates each term of the series.

Closed Form Expression

A closed form expression, when discussing series, refers to a single simplified formula that represents the sum of the terms in the series without explicitly listing them. This allows for the calculation of a series' value quickly and efficiently, even when the number of terms is very large. For example, the closed form expression for a finite geometric series is \( a \frac{1 - r^{n+1}}{1 - r} \), where \( a \) is the first term of the series, \( r \) is the common ratio, and \( n \) is the number of terms. Taking example (a) again, the series \( \Sigma_{n=2}^{18} \frac{2}{3^{2n}} \) translates to the closed form \( \frac{2}{3^{2}} \frac{1 - (\frac{1}{9})^{18-2+1}}{1 - \frac{1}{9}} \). Understanding how to derive and apply closed form expressions is a fundamental aspect of working with series, as it allows one to bypass the painstaking process of adding a potentially large number of terms.

Series Convergence

Series convergence is a fundamental concept when dealing with infinite series. A series converges if the sum of its terms approaches a fixed number as the number of terms increases towards infinity. Conversely, if the sum grows without bound or oscillates indefinitely as more terms are added, the series is said to diverge. The convergence of a geometric series depends on the absolute value of the common ratio, \( |r| \); if \( |r| < 1 \) the series converges, otherwise it diverges. For instance, in part (a) of our exercise, the common ratio is \( \frac{1}{9} \), which is less than 1, hence the series converges. However, if given an infinite series, determining convergence requires analyzing the behavior of the terms as they progress towards infinity, often using tests such as the Ratio Test, the Root Test, or comparison with known convergent series.