Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges. (a) \(-\frac{4}{3}-\frac{1}{2}-\frac{3}{16}-\frac{9}{128}+\cdots\) (b) \(-\frac{1}{100}+\frac{1.1}{(100)^{2}}-\frac{1.21}{(100)^{3}}+\frac{1.331}{(100)^{4}}-\cdots\) (c) \(-\frac{7}{10000}+\frac{7}{11000}-\frac{7}{12100}+\frac{7}{13310}-\cdots\) (d) \(1-x+x^{2}-x^{3}+\cdots\) for \(|x|<1\)

Step-by-Step Solution

Answer

(a) The series diverges. (b) The series converges to -0.01. (c) The series converges to -0.0007. (d) The series converges to 1/(1+x).

Step 1: Identify Each Series' First Term and Common Ratio

Identify the first term and the common ratio for each series by dividing the second term by the first term, the third term by the second term, etc., and check if the ratio is the same for each consecutive pair. If this ratio varies, the series isn't geometric and you'll have to stop here since the following steps only apply to geometric series.

Step 2: Determine If The Series Converge or Diverge

If the common ratio \( r \) is between -1 and 1, exclusive, then the series converges. If \( r \) falls outside that range, the series diverges. Check that the ratio \( r \) found for each series in Step 1 falls within this range.

Step 3: Calculate the Limit Of The Series for Converging Series

For each series that converges, calculate the limit that the series converge to using the formula \( S = \frac{{a}}{1-r}\) where \( a \) is the first term of the series and \( r \) is the common ratio.

Step 4: Answer The Exercise

State whether each series converges or diverges. If it converges, specify the limit that it converges to.

Key Concepts

Common Ratio

In a geometric series, the common ratio is a crucial factor. It determines the pattern of the sequence. If you want to find out whether a series is geometric, start by calculating the common ratio.

• Identifying the Common Ratio:To do this, simply divide the second term by the first term, third by the second, and so on. If the quotient remains the same each time, this constant value is your common ratio, denoted as \( r \).
• Example from Exercise:Take series (a) \(-\frac{4}{3} - \frac{1}{2} - \frac{3}{16} - \frac{9}{128} + \cdots\). Here, dividing \(-\frac{1}{2}\) by \(-\frac{4}{3}\), you get \( r = \frac{3}{8} \).

The common ratio informs whether a geometric series will converge or diverge.

Convergence and Divergence

To determine whether a geometric series converges or diverges, understanding the common ratio helps.

• Convergence: A geometric series converges if its common ratio \( r \) is between \(-1\) and \(1\) (exclusive). This means the terms of the series get progressively smaller in absolute value.
• Divergence:If the common ratio \( r \) is outside this range, the series diverges, meaning the sum increases indefinitely.
• Practical Example:Let's look at series (b) from the exercise:\(-\frac{1}{100} + \frac{1.1}{(100)^{2}} - \frac{1.21}{(100)^{3}} + \frac{1.331}{(100)^{4}} - \cdots \) Analysing the ratio, we get \( r = -1.1 \cdot \frac{1}{100} \). Since \(|r| = 0.01\), which is less than 1, the series converges.

By determining convergence or divergence, you can predict the future behavior of the series.

Series Limit Calculation

If your geometric series converges, the next step is to calculate its limit. This limit represents the value that the series approaches as the number of terms goes to infinity.

• Formula:For a converging series, use the formula \[ S = \frac{a}{1 - r} \] where \( a \) is the first term, and \( r \) is the common ratio.
• Example:Let's consider series (d) from the exercise:\( 1-x+x^{2}-x^{3}+\cdots \) (for \(|x|<1\)). The first term \( a = 1 \) and \( r = -x \) With this, the sum becomes: \[ S = \frac{1}{1 - (-x)} = \frac{1}{1 + x} \]

Calculating the limit helps you understand what value the entire series will settle at if it sums infinitely many terms.