Problem 10 - Applied Calculus

Question

Find the sum, if it exists. $$ 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^{10}} $$

Step-by-Step Solution

Answer

The sum of the sequence is approximately 5.997.

Step 1: Identify the Sequence

First, let's identify the kind of sequence we have. The sequence begins with 3 and each subsequent term is half of the previous one, except the first term is 3 multiplied by these fractions: \[ 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots \] This sequence is a **geometric sequence**.

Step 2: Determine the First Term and Ratio

For a geometric sequence, the general form is \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the first term and \( r \) is the common ratio. Here, the first term \( a = 3 \), and the common ratio \( r = \frac{1}{2} \).

Step 3: Apply the Formula for the Sum of a Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a \frac{1-r^n}{1-r} \] In this sequence, \( a = 3 \), \( r = \frac{1}{2} \), and \( n = 11 \) (since there are 11 terms: \( 3 \), \( \frac{3}{2}, \ldots, \frac{3}{2^{10}} \)).

Step 4: Calculate the Sum

Plug the numbers into the formula: \[ S_{11} = 3 \frac{1-(\frac{1}{2})^{11}}{1-\frac{1}{2}} \] Calculate \( (\frac{1}{2})^{11} = \frac{1}{2048} \) and substitute: \[ S_{11} = 3 \frac{1-\frac{1}{2048}}{\frac{1}{2}} \] Simplify: \[ S_{11} = 3 \cdot 2 \cdot \left(1 - \frac{1}{2048} \right) = 6 \left(1 - \frac{1}{2048}\right) \] \[ S_{11} = 6 \left( \frac{2047}{2048} \right) = \frac{12282}{2048} = 6 \times 0.99951171875 \] \[ S_{11} \approx 5.997 \]

Step 5: Conclude the Result

Thus, the sum of the sequence \( 3 + \frac{3}{2} + \frac{3}{4} + \cdots + \frac{3}{2^{10}} \) is approximately \( 5.997 \).

Key Concepts

Sum of Geometric Series

A geometric series is a sum of the terms in a geometric sequence. To find the sum of a geometric series, especially when the sequence is finite, we use a specific formula. This formula is:

• \[ S_n = a \frac{1-r^n}{1-r} \]

Here, \( S_n \) represents the sum of the first \( n \) terms, \( a \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the total number of terms.
It's important to note that if the absolute value of \( r \) is less than 1 and the sequence continues indefinitely, an infinite geometric series will converge to a finite sum, given by this simpler version of the formula:

• \[ S = \frac{a}{1 - r} \]

However, our series in this exercise has a fixed number of terms, which is why we use the finite series formula. The calculation simplifies the expression and allows us to determine the sum precisely. In this case, using the formula and plugging in our values, we compute the sum as approximately 5.997.

Common Ratio

The common ratio in a geometric sequence is a crucial part of the sequence's structure. It determines the factor by which we multiply each term to get to the next one. In our exercise, the common ratio is given as \( r = \frac{1}{2} \). This ratio tells us that each term in the sequence is half of the preceding term.
To find the common ratio, simply divide any term by the previous term. For example, if the first term is \( 3 \) and the second term is \( \frac{3}{2} \), then

• \[ r = \frac{3/2}{3} = \frac{1}{2} \]

Identifying the common ratio helps us not only understand the growth pattern but also enables us to apply the formula for the sum of the series effectively.
A sequence with a common ratio less than 1, like our example here, usually results in a decreasing sequence, heading towards smaller and smaller numbers.

First Term

In any geometric progression, the first term sets the stage for the rest of the sequence. It is denoted by \( a \) and represents the starting point of the sequence.
In our given series, the first term \( a \) is \( 3 \). This gives us a base from which all subsequent terms are generated by multiplying by the common ratio.
Understanding the first term is fundamental because it influences every term in the sequence. When calculating the sum of a geometric series, this first term is part of the multiplier that determines the impact each subsequent term has on the overall sum.
This notion is woven into the sum formula, where \( a \) is used to scale the result from the application of the common ratio over multiple terms.

Geometric Progression

A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
An example of a geometric progression can be seen in our exercise where the sequence starts with 3 and each following term is halved, creating the series:

• \( 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots \)

Geometric progressions have various applications in mathematics and real-world scenarios, explaining phenomena that involve exponential growth or decay, such as population growth, financial interest rates, and more.
In a mathematical context, recognizing a series as a geometric progression allows us to use specific formulas and techniques to solve problems, such as finding the sum or predicting further terms. Understanding these sequences and their principles forms a core foundation for exploring more advanced mathematical concepts.