Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n}} $$

Step-by-Step Solution

Answer

The first two terms of the series are -1/3 and 1/9. The series converges because the absolute value of the ratio is less than 1. The sum of the series is -1/2.

Step 1: Write out the first two terms of the series

To find the first two terms n is substituted by 1 and 2 respectively. For n=1, the term will be \( \frac{(-1)^1}{3^1} = -\frac{1}{3} \). For n=2, the term will be \( \frac{(-1)^2}{3^2} = \frac{1}{9} \). So, the first two terms are -1/3 and 1/9.

Step 2: Determine whether the series converges or not

The given series is an infinite geometric series, and a geometric series converges if the absolute value of the ratio is less than 1. The ratio r is \( \frac{(-1)}{3} \), and |r| is \( \frac{1}{3} \). So since \( \frac{1}{3} < 1 \), the series converges.

Step 3: Determine the sum of the series

Since the series converges, we can find the sum using the formula for an infinite geometric series: \( S = \frac{a}{1 - r} \) where a is the first term of the series and r is the common ratio. In this case, a = -1/3 and r = \( \frac{(-1)}{3} \). Substituting the values, the sum of the series is -1/2.

Key Concepts

Series Convergence

Understanding the concept of series convergence is essential when dealing with infinite series. A series is essentially a summer up of infinitely many terms, and whether or not it converges determines if we can assign a finite value to this sum.

For a series to converge, the terms must approach zero as n approaches infinity, which means that with each new term added, the overall sum gets closer to a certain number. If the terms do not approach zero, or if they grow larger, the series diverges and has no finite sum.

In our exercise, we discussed the convergence of an infinite geometric series. Such a series converges if the absolute value of the common ratio, denoted as |r|, is less than 1. For the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n}} \), we verify that this condition holds as |r| equals \( \frac{1}{3} \) which is less than 1. Consequently, we confirm that the series converges.

Geometric Series Sum

One of the most powerful tools in calculus is the formula for finding the sum of an infinite geometric series. This formula only applies if the series converges, as explained in the previous section.

The sum of an infinite geometric series is given by the simple formula: \( S = \frac{a}{1 - r} \), where \( a \) is the first term of the series and \( r \) is the common ratio between consecutive terms. This formula is remarkable because it can convert an infinitely long process into a single, finite expression.

Using our exercise as an example, we applied this formula to find that the sum of our series is \( -1/2 \). This was obtained by recognizing that the first term \( a \) is \( -1/3 \) and the ratio \( r \) is \( -1/3 \), thus allowing us to calculate the sum using the aforementioned formula.

Calculus Problems and Solutions

Problems involving infinite series are frequent in calculus, and they present special challenges due to their infinite nature. To address these challenges, specific techniques and solutions have been developed, such as the convergence tests for series and the sum formulas for convergent geometric series.

Approaching these problems methodically is key. First, identify the type of series; second, apply relevant tests or formulas to solve for convergence and sum; and third, interpret results in the context of the problem.

In the step-by-step solution we provided, we applied the principles and computations systematically to find the required sum. Understanding these methods allow students not only to solve the problems but also to grasp the concept of infinity in mathematical terms.

Rate of Convergence

The rate of convergence refers to how quickly a series approaches its limit as the terms are added. This is particularly relevant when discussing infinite series, as some series converge faster than others.

A geometric series with a smaller absolute ratio will converge more quickly, as each term decreases in magnitude at a faster rate than a series with a larger ratio. The ratio's proximity to zero is directly correlated to the speed of convergence.

In our exercise, the rate of convergence can be considered relatively quick since the absolute value of the ratio is \( \frac{1}{3} \), which decreases the term size substantially with each step. This concept is crucial when we work with infinite series in practical applications, where determining the rate of convergence can inform us about the efficiency and viability of using series approximations for computations.