Problem 127 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. \(\ln 2+\ln 3+\ln 4+\ln 5\)

Step-by-Step Solution

Answer

The sum \( \ln 2 + \ln 3 + \ln 4 + \ln 5 \) in sigma notation is \( \sum_{i=2}^{5} \ln i \).

Step 1: Identify the Pattern in the Sum

The problem asks us to express the sum \( \ln 2 + \ln 3 + \ln 4 + \ln 5 \) in sigma notation. We first identify the sequence in the sum: \( \ln 2, \ln 3, \ln 4, \ln 5 \). Here, the natural logarithms have consecutive integers starting from 2 up to 5.

Step 2: Establish the Sigma Notation Variables

To write in sigma notation, we need to identify the lower and upper bounds of the sum. The sequence of numbers starts at 2 and ends at 5. Therefore, the lower bound (starting point) \( i = 2 \) and the upper bound (ending point) is \( i = 5 \).

Step 3: Formulate the Sigma Notation Expression

The general term for this series is based on the identified pattern, which is \( \ln i \), where \( i \) is the running index from 2 to 5. We can now write the series in sigma notation: \[ \sum_{i=2}^{5} \ln i \].

Key Concepts

Sequences and Series

When dealing with sequences and series, it's helpful to understand that they represent collections of numbers listed in a specific order. In this context, a **sequence** is simply an ordered list of numbers; for example, the sequence 2, 3, 4, and 5 is a list of consecutive integers. Meanwhile, a **series** refers to the sum of the elements of a sequence. For instance, \( 2 + 3 + 4 + 5\) is a series derived from the sequence above.

In mathematical problems involving sums, we often see patterns where the numbers are expressed using a consistent rule or formula, such as an arithmetic or geometric progression. For the original exercise, the series is defined by the natural logarithm function applied to each element of the sequence, \( \ ext{ln}2, \text{ln}3, \text{ln}4, \text{ln}5\). Being able to see these patterns helps in translating a series into sigma notation, which is a compact way to express the sum.

Natural Logarithms

**Natural logarithms** are logarithms with the base of \(e\), an irrational constant approximately equal to 2.71828. They are written as \( \text{ln} \, x \), representing the power to which \( e \) must be raised to obtain the value \( x \).

Natural logarithms have several useful properties that simplify complex mathematical expressions:

• The Product Rule: \( \ln(a \times b) = \ln a + \ln b \)
• The Quotient Rule: \( \ln(\frac{a}{b}) = \ln a - \ln b \)
• The Power Rule: \( \ln(a^b) = b \times \ln a \)

In this context, we're not directly using these properties, but understanding them helps in recognizing why the sum of natural logarithms can be a valuable form.

Mathematical Notation

Mathematical notation is like a special language used to express and communicate complex ideas in a simple and efficient way. Sigma notation is one of these powerful tools. It allows us to write long sums in a compact form.

In sigma notation, the symbol \( \Sigma \) (the Greek letter Sigma) is used to denote a sum. The notation looks like this: \( \sum_{i=a}^{b} f(i) \), where:

• \( \sum \) is the sigma symbol indicating summation.
• \( i \) is the index of summation, representing each number in the series.
• \( a \) is the lower limit of summation, marking the start of the sequence.
• \( b \) is the upper limit of summation, marking the end of the sequence.
• \( f(i) \) is the function applied to the elements being summed.

In the exercise, we concluded that the sum \( \ln 2 + \ln 3 + \ln 4 + \ln 5 \) can be written using sigma notation as \( \sum_{i=2}^{5} \ln i\). This efficient representation is vital in understanding and solving problems involving large series.