Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the sum of the following. (If there is no nite sum, say so.) (a) \(3+9+27+\cdots+3^{20}\) (b) \(\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+\cdots+\left(\frac{2}{3}\right)^{n}+\cdots\) (c) \((0.2)(10)+(0.2)(100)+(0.2)(1000)+\cdots\) (d) \(3+3(0.8)+3(0.8)^{2}+3(0.8)^{3}+\cdots\) (e) \((0.2)+(0.2)(1.3)+(0.2)(1.3)^{2}+(0.2)(1.3)^{3}+\cdots\) (f) \(1+x^{2}+x^{4}+x^{6}+\cdots\) for \(-1

Step-by-Step Solution

Answer

The solutions are \(S_a = 3^{20} - 3\), \(S_b = 2\), \(S_c\) is undefined, \(S_d = 15\), \(S_e\) is undefined, and \(S_f = \frac{1}{1 - x^2}\).

Step 1: Solve (a)

First, let's consider the geometric series \(3 + 9 + 27 + \dots + 3^{20}\). In this case, our first term \(a = 3\), our common ratio \(r = 3\), and there are \(n = 20\) terms. Hence, we use the formula for the sum of a finite geometric series, \( S = \frac{a(1 - r^n)}{1 - r} \), and get \( S_a = \frac{3(1 - 3^{20})}{1 - 3} = 3^{20} - 3 \)

Step 2: Solve (b)

Now, let's consider the geometric series \(\frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \dots\). Our first term is \(a = \frac{2}{3}\) and our common ratio is \(r = \frac{2}{3}\). Since this is an infinite series, we use the formula for the sum of an infinite geometric series (which applies because \(\left| \frac{2}{3} \right| < 1)\), \(S = \frac{a}{1 - r}\), and get \(S_b = \frac{\frac{2}{3}}{1 - \frac{2}{3}} = 2\)

Step 3: Solve (c)

For the geometric series \(0.2(10) + 0.2(100) + 0.2(1000) + \dots\), our first term is \(a = 0.2(10) = 2\) and the common ratio is \(r = \frac{0.2(100)}{0.2(10)} = 10\). This therefore forms an infinite geometric series with \(r > 1\); hence, it does not have a finite sum, indicating that \(S_c\) is undefined.

Step 4: Solve (d)

Now, let's consider the geometric series \(3+3(0.8)+3(0.8)^{2}+3(0.8)^{3}+\cdots\). Our first term is \(a = 3\), and our common ratio is \(r = 0.8\). Since this is an infinite series, we use the formula for the sum of an infinite geometric series (which applies because \(|0.8| < 1)\), \(S = \frac{a}{1 - r}\), and get \(S_d = \frac{3}{1 - 0.8} = 15\)

Step 5: Solve (e)

Now, let's look at the geometric series \((0.2)+(0.2)(1.3)+(0.2)(1.3)^{2}+(0.2)(1.3)^{3}+\cdots\). Our first term is \(a = 0.2\), and our common ratio is \(r = 1.3\). This series is infinite, but since the common ratio is greater than 1, we can't use the formula for the sum of an infinite geometric series. Hence, \(S_e\) is undefined.

Step 6: Solve (f)

For the last series, \(1+x^{2}+x^{4}+x^{6}+\cdots\) for \(-1

Key Concepts

finite sum

A finite sum in the context of a geometric series involves summing a sequence of terms that eventually comes to an end. Each term in a geometric series is derived by multiplying the previous term by a constant known as the common ratio. To calculate the sum of a finite geometric series, we use the formula:

• \( S = \frac{a(1 - r^n)}{1 - r} \)

where:

• \( a \) is the first term,
• \( r \) is the common ratio,
• \( n \) is the number of terms.

An example from the exercise involves the series \(3 + 9 + 27 + \cdots + 3^{20}\), where the finite sum is calculated up to \(3^{20}\). By plugging the values into our formula, we find that its finite sum is \(3^{20} - 3\). This demonstrates the power of formulas in efficiently calculating such sums.

infinite series

An infinite series is a sequence of numbers that continues indefinitely. Unlike finite series, infinite series don't stop at a certain number of terms. In geometry, if the absolute value of the common ratio \( |r| < 1 \), these series can actually sum up to a finite value. For finding the sum of an infinite geometric series, the formula used is:

• \( S = \frac{a}{1 - r} \)

However, this formula is only applicable when the series converges, which happens when \(|r|<1\). For instance, the series \( \frac{2}{3} + (\frac{2}{3})^2 + \cdots \) meets this criterion with a sum equal to \(2\), as the common ratio is less than 1.

common ratio

The common ratio in a geometric series is the constant factor between consecutive terms. It is a crucial element that defines the type of progression depicted by the series. Calculating the common ratio is straightforward:

• Divide the second term by the first term.

For example, when examining the series \(3 + 9 + 27 + \cdots + 3^{20}\), dividing the second term (9) by the first term (3), we find the common ratio to be \(3\).
Understanding the common ratio is vital not only for determining the progression but also for applying the right formula to find sums, whether finite or infinite. It's important to ensure that the common ratio remains consistent across the series. If it fluctuates, the series is not geometric.

sum of a geometric series

The sum of a geometric series depends on whether it is finite or infinite. For finite series, the sum is found using:

• \( S = \frac{a(1 - r^n)}{1 - r} \)

while for infinite series, when \(|r|<1\), the formula is:

• \( S = \frac{a}{1 - r} \)

In the exercise solution, knowing how to apply these formulas reveals whether we can compute a sum or not. If the series is of infinite length and the common ratio \(|r| \geq 1\), then it diverges, making it impossible to find a finite sum, as in the case of the series with terms like \(0.2(10) + 0.2(100) + \cdots\).
These mathematical tools provide a structured way to approach and understand series, ensuring that even complex sequences can be distilled into manageable computations.

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Problem 11

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Problem 12

Other exercises in this chapter

Problem 11

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) "

View solution

Problem 11

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you are saving for a big trip abroad. You estimate that you 11 nee

View solution

Problem 12

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) "

View solution

Problem 12

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you are saving to buy some cattle. You plan to put \(\$ 200\) into

View solution

Key Concepts

finite sum

A finite sum in the context of a geometric series involves summing a sequence of terms that eventually comes to an end. Each term in a geometric series is derived by multiplying the previous term by a constant known as the common ratio. To calculate the sum of a finite geometric series, we use the formula:

• \( S = \frac{a(1 - r^n)}{1 - r} \)

where:

• \( a \) is the first term,
• \( r \) is the common ratio,
• \( n \) is the number of terms.

An example from the exercise involves the series \(3 + 9 + 27 + \cdots + 3^{20}\), where the finite sum is calculated up to \(3^{20}\). By plugging the values into our formula, we find that its finite sum is \(3^{20} - 3\). This demonstrates the power of formulas in efficiently calculating such sums.

infinite series

An infinite series is a sequence of numbers that continues indefinitely. Unlike finite series, infinite series don't stop at a certain number of terms. In geometry, if the absolute value of the common ratio \( |r| < 1 \), these series can actually sum up to a finite value. For finding the sum of an infinite geometric series, the formula used is:

• \( S = \frac{a}{1 - r} \)

However, this formula is only applicable when the series converges, which happens when \(|r|<1\). For instance, the series \( \frac{2}{3} + (\frac{2}{3})^2 + \cdots \) meets this criterion with a sum equal to \(2\), as the common ratio is less than 1.

common ratio

The common ratio in a geometric series is the constant factor between consecutive terms. It is a crucial element that defines the type of progression depicted by the series. Calculating the common ratio is straightforward:

• Divide the second term by the first term.

For example, when examining the series \(3 + 9 + 27 + \cdots + 3^{20}\), dividing the second term (9) by the first term (3), we find the common ratio to be \(3\).
Understanding the common ratio is vital not only for determining the progression but also for applying the right formula to find sums, whether finite or infinite. It's important to ensure that the common ratio remains consistent across the series. If it fluctuates, the series is not geometric.

sum of a geometric series

The sum of a geometric series depends on whether it is finite or infinite. For finite series, the sum is found using:

• \( S = \frac{a(1 - r^n)}{1 - r} \)

while for infinite series, when \(|r|<1\), the formula is:

• \( S = \frac{a}{1 - r} \)

In the exercise solution, knowing how to apply these formulas reveals whether we can compute a sum or not. If the series is of infinite length and the common ratio \(|r| \geq 1\), then it diverges, making it impossible to find a finite sum, as in the case of the series with terms like \(0.2(10) + 0.2(100) + \cdots\).
These mathematical tools provide a structured way to approach and understand series, ensuring that even complex sequences can be distilled into manageable computations.