Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

$\sum_{k=1}^{6} \frac{3}{k+1} $$

Step-by-Step Solution

Answer

\( \frac{3}{2} + \frac{3}{3} + \frac{3}{4} + \frac{3}{5} + \frac{3}{6} + \frac{3}{7} \) is the summation of the given series.

Step 1: Understand the Sigma Notation

The expression \( \sum_{k=1}^{6} \frac{3}{k+1} \) is a summation notation. It indicates that we need to substitute each integer value of \(k\) from 1 to 6 into the expression \( \frac{3}{k+1} \) and add all the results together.

Step 2: Substitute Values and Calculate Each Term

Substitute each integer value of \( k \) from 1 to 6 into the expression \( \frac{3}{k+1} \). Calculate each term individually:- For \( k = 1 \): \( \frac{3}{1+1} = \frac{3}{2} \)- For \( k = 2 \): \( \frac{3}{2+1} = \frac{3}{3} \)- For \( k = 3 \): \( \frac{3}{3+1} = \frac{3}{4} \)- For \( k = 4 \): \( \frac{3}{4+1} = \frac{3}{5} \)- For \( k = 5 \): \( \frac{3}{5+1} = \frac{3}{6} \)- For \( k = 6 \): \( \frac{3}{6+1} = \frac{3}{7} \)

Step 3: Sum All the Calculated Terms

Add all the calculated terms together:\[ \frac{3}{2} + \frac{3}{3} + \frac{3}{4} + \frac{3}{5} + \frac{3}{6} + \frac{3}{7} \]

Key Concepts

Summation

Summation is a mathematical concept that involves adding a sequence of numbers or expressions together. It is often represented by the Greek letter sigma (\( \Sigma \)). In the exercise given, the summation notation \( \sum_{k=1}^{6} \frac{3}{k+1} \) indicates that we will start with \( k = 1 \) and increment \( k \) by 1 until it reaches 6. For each value of \( k \), the corresponding term \( \frac{3}{k+1} \) is calculated.

• The lower limit of the summation (1) is where we start substituting the values in place of \( k \).
• The upper limit (6) tells us when to stop.
• The expression \( \frac{3}{k+1} \) is the function of \( k \) that we evaluate for each integer between these limits.

Understanding summation is crucial as it simplifies the process of addition by using a compact notation to represent a series of terms. It can make seemingly complex calculations more manageable by breaking them into parts.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. In our example, the expression \( \frac{3}{k+1} \) is a rational expression where \(3\) is the numerator, and \(k+1\) is the denominator. Understanding rational expressions involves:

• Recognizing that they can be simplified by factoring and reducing common terms.
• Being able to substitute specific values into the variable present, which in this case, is the integer values \( k = 1, 2, 3, 4, 5, \) and \(6\).
• Knowing that the denominator must never be zero, as division by zero is undefined, but this doesn't apply here since no value of \( k \) from 1 to 6 makes \( k+1 \) zero.

When dealing with rational expressions, calculating each term by substituting values directly gives a streamlined approach to solving such summations.

Finite Series

A finite series is the sum of a finite number of terms. In other words, it's a summation where both the lower and upper limits are specified, which means we only add a set number of terms together. In our exercise, the finite series is \( \frac{3}{2} + \frac{3}{3} + \frac{3}{4} + \frac{3}{5} + \frac{3}{6} + \frac{3}{7} \).The finite nature of this series refers to:

• There are only six terms to compute and add together, as defined by the limits of our summation.
• The end result is a specific numerical value that represents the sum of these calculated terms.
• Finite series are often easier to work with than infinite ones because they have clear boundaries, allowing precise computations.

Understanding finite series helps in performing exact calculations efficiently. The method relies on substitution, followed by addition of evaluated terms, leading to a total sum that characterizes the series.