Problem 3 - Calculus

Question

Find the value of the indicated sum. \(\sum_{k=1}^{7} \frac{1}{k+1}\)

Step-by-Step Solution

Answer

The sum is approximately 1.7179.

Step 1: Identify the Expression to be Summed

The given sum to be calculated is \( \sum_{k=1}^{7} \frac{1}{k+1} \). We need to find the sum of each term where the expression is of the form \( \frac{1}{k+1} \) and \( k \) varies from 1 to 7.

Step 2: List Each Term in the Series

Substitute each integer value of \( k \) from 1 to 7 into the expression \( \frac{1}{k+1} \) to get the individual terms:- For \( k = 1 \): \( \frac{1}{1+1} = \frac{1}{2} \)- For \( k = 2 \): \( \frac{1}{2+1} = \frac{1}{3} \)- For \( k = 3 \): \( \frac{1}{3+1} = \frac{1}{4} \)- For \( k = 4 \): \( \frac{1}{4+1} = \frac{1}{5} \)- For \( k = 5 \): \( \frac{1}{5+1} = \frac{1}{6} \)- For \( k = 6 \): \( \frac{1}{6+1} = \frac{1}{7} \)- For \( k = 7 \): \( \frac{1}{7+1} = \frac{1}{8} \)

Step 3: Calculate the Sum of the Series

Add all the terms calculated in the previous step:\[\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\]Using a calculator or reducing each fraction to a common denominator to simplify the addition, you would find that:\[\approx 1.7179\] when summed together.

Key Concepts

Series

A series is essentially the operation of adding a sequence of numbers. In mathematics, this is often presented with the sigma notation, which signifies summing multiple terms that are produced by a particular expression. A series can be finite or infinite based on how many terms are added. While an infinite series continues indefinitely, a finite series, like our example, ends after a certain number of terms.
Series come in various types, and one common form is the arithmetic series, where each term in the sequence is derived by adding a fixed number to the previous term.
The series we are looking at involves the sum of fractions, and to solve it, you need to identify each term of the series separately before adding them together.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant to the previous term. This constant is known as the common difference. For example, the list 2, 4, 6, 8 is an arithmetic sequence with a common difference of 2.

• First Term: The starting number of the sequence.
• Common Difference: The amount added to each term to get the next.

This differs from our example problem because the sequence in the problem isn't formed by a constant addition; it involves a reciprocal rule. Understanding the concept of an arithmetic sequence helps appreciate the systematic nature of many types of sequences and their behavior.

Mathematical Notation

Mathematical notation uses symbols and abbreviations to convey mathematical ideas succinctly. In our exercise, tools like the sigma notation come into play. Sigma notation, written as \( \sum \), tells you to add up all terms from a given sequence produced by a specific rule.

For example, in \( \sum_{k=1}^{7} \frac{1}{k+1} \), sigma indicates summation of terms like \( \frac{1}{2}, \frac{1}{3}, \dots \frac{1}{8}\). The expression following the sigma symbol must define how each term is calculated. The limits \(k=1\) and \(k=7\) signify where the sequence starts and stops.

• Upper Limit: Defines the last number in the sequence.
• Lower Limit: Defines the first number in the sequence.

Understanding this notation is key to evaluating series and sequences efficiently.