Problem 33 - Precalculus: Mathematical for Calculus

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence.. $$\frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \frac{1}{3^{4}}, \dots$$

Step-by-Step Solution

Answer

The first six partial sums are \( S_1 = \frac{1}{3}, S_2 = \frac{4}{9}, S_3 = \frac{13}{27}, S_4 = \frac{40}{81}, S_5 = \frac{121}{243}, S_6 = \frac{364}{729} \).

Step 1: Identify the Sequence Terms

The given sequence is: \( \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, \dots \). We need to identify the terms from the sequence to use them in calculating partial sums.

Step 2: Calculate the First Partial Sum \( S_1 \)

The first partial sum \( S_1 \) is simply the first term of the sequence: \( S_1 = \frac{1}{3} \).

Step 3: Calculate the Second Partial Sum \( S_2 \)

Add the first and the second terms of the sequence: \( S_2 = \frac{1}{3} + \frac{1}{3^2} = \frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9} \).

Step 4: Calculate the Third Partial Sum \( S_3 \)

Add the first three terms of the sequence: \( S_3 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{9}{27} + \frac{3}{27} + \frac{1}{27} = \frac{13}{27} \).

Step 5: Calculate the Fourth Partial Sum \( S_4 \)

Add the first four terms of the sequence: \( S_4 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{27}{81} + \frac{9}{81} + \frac{3}{81} + \frac{1}{81} = \frac{40}{81} \).

Step 6: Calculate the Fifth Partial Sum \( S_5 \)

Add the first five terms of the sequence: \( S_5 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} = \frac{81}{243} + \frac{27}{243} + \frac{9}{243} + \frac{3}{243} + \frac{1}{243} = \frac{121}{243} \).

Step 7: Calculate the Sixth Partial Sum \( S_6 \)

Add the first six terms of the sequence: \( S_6 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729} = \frac{243}{729} + \frac{81}{729} + \frac{27}{729} + \frac{9}{729} + \frac{3}{729} + \frac{1}{729} = \frac{364}{729} \).

Key Concepts

Sequence

A sequence is an ordered list of numbers. Each number in the list is called a term. In mathematics, sequences can be finite or infinite and often follow a specific rule or pattern.
For instance, the sequence given in the exercise is:

• \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \dots \)

This sequence is infinite, meaning it continues indefinitely. The rule guiding this sequence is that each term is one-third of the preceding term. Understanding sequences is crucial in studying calculus and algebra since they form the basis for series and other mathematical concepts. The sequence given in this example is also geometric, meaning that each term is derived by multiplying the previous term by a fixed, non-zero number called the common ratio, which in this case is \( \frac{1}{3} \).
Sequences help us understand patterns and are used in various fields such as finance, computer science, and physics.

Geometric Series

A geometric series is a sum of the terms of a geometric sequence. Each successive term is obtained by multiplying the previous term by a constant called the common ratio. In our exercise, the sequence is geometric with the common ratio, \( r = \frac{1}{3} \).

• A typical geometric series looks like this: \( a + ar + ar^2 + ar^3 + \dots \)
• Where \( a \) is the first term, and \( r \) is the common ratio.

The series explored here begins with the term \( a = \frac{1}{3} \), making its partial sums a fascinating aspect to study. The nth partial sum is the total obtained by adding the first n terms of a sequence.
For example:

• The first partial sum \( S_1 \) is just \( \frac{1}{3} \).
• The second partial sum \( S_2 = \frac{1}{3} + \frac{1}{9} \).

As you see above, the pattern continues where each term is calculated based on the geometric rule. Understanding geometric series is vital because they appear in many real-world applications, such as calculating compound interest or analyzing animal reproduction patterns.

Series Convergence

Series convergence refers to whether a series approaches a finite value as more terms are added. In mathematical terms, a series converges if its partial sums approach a specific number as the number of terms increases indefinitely.
In our geometric series with common ratio \( r = \frac{1}{3} \), we examine convergence by looking at whether the series' partial sums stabilize around a certain value.

• If \( |r| < 1 \), the series converges to the sum \( \frac{a}{1-r} \).
• If \( |r| \geq 1 \), the series does not converge, meaning it grows indefinitely.

Since \( \frac{1}{3} \) is less than 1, the series converges. In our case, given that the common ratio is \( \frac{1}{3} \) and starting term \( a = \frac{1}{3} \), the series converges to a sum: \[ S = \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = \frac{1}{2} \].Understanding series convergence is essential in calculus and mathematical analysis, as it provides insights into the behavior of infinite processes and systems.