Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Write the sum using summation notation. (a) \(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots\) (b) \(\frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots\)

Step-by-Step Solution

Answer

(a) The summation notation for the series is \(\sum_{n=1}^{\infty} \frac{2}{3^{n}}\). (b) The summation notation for the series is \(\sum_{n=-1}^{\infty} 2(3^{n})\)

Step 1: Observe the Pattern in Series (a)

In the series \(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots\), it can be observed that each term is a fraction where the numerator is constant, 2 and the denominator is a power of 3.

Step 2: Formulate the General Term (Series a)

Based on the pattern observed in the series, the general term for (a) can be expressed as \(\frac{2}{3^{n}}\), where \(n\) starts at 1 and increases by 1.

Step 3: Write the Summation Notation (Series a)

Using the sigma notation for series, the series (a) can be represented as \(\sum_{n=1}^{\infty} \frac{2}{3^{n}}\)

Step 4: Observe the Pattern in Series (b)

In the series \(\frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots\), it can be observed that each term is increasing power of 3 starting at negative 1 and the entire term is multiplied by 2.

Step 5: Formulate the General Term (Series b)

Based on the pattern observed in the series, the general term for (b) can be expressed as \(2(3^{n})\), where \(n\) starts at -1 and increases by 1.

Step 6: Write the Summation Notation (Series b)

Using the sigma notation for series, the series (b) can be represented as \(\sum_{n=-1}^{\infty} 2(3^{n})\)

Key Concepts

Geometric Series

A geometric series is a sum of terms in a sequence where each term is found by multiplying the previous term by a constant factor. This constant is known as the common ratio. Understanding geometric series is crucial when evaluating many types of sequences.

• For example, in series (a) from the exercise, we have \((\frac{2}{3}) + (\frac{2}{9}) + (\frac{2}{27}) + \cdots\),\ where each term is formed by multiplying the preceding term by \(\frac{1}{3}\).
• In series (b), the sequence \((\frac{2}{3}) + 2 + 6 + \cdots\) contains terms that are each a multiplied result by a power of 3, denoted as \(2(3^{n})\).
• The importance of the geometric series is its ability to model exponential growth or decay in various scenarios.

A key feature of geometric series is the fact they can sum up to a finite value if the common ratio is between -1 and 1, even if they have infinitely many terms. This is the basis for the concept called a converging series.

Infinite Series

An infinite series is a series that continues indefinitely, without stopping. This is represented mathematically by extending the upper limit of the sum to infinity. In an infinite series, one challenges understanding because the addition of infinitely many terms might not converge to a finite sum.

• In exercise (a), the notation \(\sum_{n=1}^{\infty}\frac{2}{3^n}\) signifies an infinite series due to the upper limit being infinity.
• Similarly, series (b) \(\sum_{n=-1}^{\infty} 2(3^n)\) also represents an infinite series.
• The concept of convergence is significant; it refers to the series approaching a fixed value as more terms are added.

To determine whether a series like these converges or not, it may require specific tests or analyses, particularly useful in the study of calculus and mathematical sequences.

Sigma Notation

Sigma notation is a compact and formal way to express the sum of terms. The symbol used is \(\Sigma\), which is the Greek letter sigma, standing for "sum."

• This notation highlights the starting and ending points of summation, as well as the general term to sum.
• For instance, in series (a), \(\sum_{n=1}^{\infty}\frac{2}{3^n}\), indicates adding terms starting from \(n=1\) through to infinity.
• In series (b), \(\sum_{n=-1}^{\infty} 2(3^n)\), reflects a summation starting with \(n=-1\).

Sigma notation serves as a powerful tool because it provides a clear, concise way to deal with complex sequences and sums, often encountered in mathematics and its applications.