Problem 20 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. $$ \frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots+\frac{1}{2^{n}} $$

Step-by-Step Solution

Answer

\( \sum_{k=0}^{n} \frac{1}{2^k} \)

Step 1: Identify the Pattern

Look at the series to recognize the pattern in terms of indices and the base function. In this series: \( \frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots \), each term is of the form \( \frac{1}{2^k} \) where \( k \) starts at 0 and goes up to \( n \).

Step 2: Define the General Term

Determine the general term of the sequence. Each term in the sequence is an inverse power of 2, specifically \( \frac{1}{2^k} \). The exponent \( k \) operates as the index which increases by 1 for each subsequent term, starting from 0.

Step 3: Write the Series in Sigma Notation

Use sigma notation to express the sum of the series. This is done by setting up the sum from \( k=0 \) to \( n \) of the term \( \frac{1}{2^k} \). Thus, the sigma notation is: \[ \sum_{k=0}^{n} \frac{1}{2^k} \]

Key Concepts

Geometric Series

Geometric series are sums composed of terms that form a geometric progression, where each term after the first is found by multiplying the previous term by a constant ratio (common ratio). In the series presented above, each term can be expressed in the form \( \frac{1}{2^k} \) with \( k \) being the index of the term. Here, the common ratio \( r \) is \( \frac{1}{2} \).

Geometric series are highly significant in mathematics because they have a simple form and useful convergence properties. If the ratio \( |r| < 1 \), the infinite series converges to a specific value. The sum of the first \( n+1 \) terms of a geometric series can be calculated by the formula:

• \( S_n = a \frac{1-r^{n+1}}{1-r} \)

where \( a \) is the first term of the series and \( r \) is the common ratio. In the case of the series in the exercise, understanding its structure helps you communicate complex ideas through simpler calculations.

Sequence and Series

Sequences and series are fundamental concepts in mathematics that allow us to describe lists of numbers that follow specific patterns and to find the sum of these ordered lists, respectively. A sequence is an ordered list of numbers; for example, the sequence in the exercise is \( \frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \dots \).

A series, on the other hand, is formed by taking the sum of the terms in a sequence. In terms of notation, if you are summing up the elements of a sequence, you express this as a series, often using sigma notation \( \Sigma \). This provides a compact way to write the series instead of listing each term. Sigma notation includes the general term, the index variable, the starting value, and the ending value. It is particularly helpful as it shortens lengthy calculations, making them appear neat and organized.

General Term

The general term of a sequence is an expression that represents the \( n \)-th term using a formula. This enables us to identify any term within the sequence without listing all previous terms. In the context of the exercise, the general term is \( \frac{1}{2^k} \). The general term allows one to replace \( k \) with any integer to find the corresponding term of the sequence.

Creating a general term from a sequence involves examining the pattern of numbers within the sequence. For geometric sequences, the general term can usually be written as:

• \( a_n = ar^{n-1} \)

where \( a \) is the first term, and \( r \) is the common ratio. Understanding how to derive and manipulate the general term lays the foundation for working with more complex mathematical concepts.