Problem 57 - Contemporary Precalculus

Question

Express the given sum in \(\Sigma\) notation and find the sum. $$\frac{1}{8}-\frac{2}{9}+\frac{3}{10}-\frac{4}{11}+\frac{5}{12}$$

Step-by-Step Solution

Answer

Question: Express the given sum in sigma notation and find the sum: \(\frac{1}{8}-\frac{2}{9}+\frac{3}{10}-\frac{4}{11}+\frac{5}{12}\) Answer: The sum can be expressed in sigma notation as \(\sum_{n=1}^5 (-1)^{n+1}\cdot\frac{n}{n+7}\) and the sum is equal to \(-\frac{1}{264}\).

Step 1: Recognize the pattern

Looking at the given sum, the general term should be in the form of: $$(-1)^{n+1}\cdot\frac{n}{n+7}$$ The terms alternate signs as denoted by the \((-1)^{n+1}\) term, and as the numerator and denominator increase by 1. **Step 2: Express the sum in sigma notation**

Step 2: Write the sum using \(\Sigma\)

Since we have identified the general term, we can now express the sum using sigma notation. The given sum has 5 terms, so the expression can be written as: $$\sum_{n=1}^5 (-1)^{n+1}\cdot\frac{n}{n+7}$$ **Step 3: Calculate the sum**

Step 3: Sum the general terms

Now, we simply need to calculate the sum by substituting the values of n in the general term expression, and adding them up: $$\sum_{n=1}^5 (-1)^{n+1}\cdot\frac{n}{n+7} = (-1)^2\cdot\frac{1}{1+7} + (-1)^3\cdot\frac{2}{2+7} + (-1)^4\cdot\frac{3}{3+7} + (-1)^5\cdot\frac{4}{4+7} + (-1)^6\cdot\frac{5}{5+7}$$ Simplify the expressions within the sum to get: $$\frac{1}{8}-\frac{2}{9}+\frac{3}{10}-\frac{4}{11}+\frac{5}{12}$$ The sum of the series is equal to the given expression: $$\frac{1}{8}-\frac{2}{9}+\frac{3}{10}-\frac{4}{11}+\frac{5}{12} = -\frac{1}{264}$

Key Concepts

Series and Sequences

In mathematics, sequences and series play crucial roles in understanding patterns and summations. A **sequence** is a list of numbers in a particular order. Each number in a sequence is called a term. Sequences can be finite or infinite, depending on whether they end or continue indefinitely.

On the other hand, a **series** is the sum of the terms in a sequence. When you see a sequence and imagine adding all its terms together, you are considering the series. For example, if you take the sequence 1, 2, 3, ..., its series would be the sum 1 + 2 + 3 + ... .

Understanding sequences and series allows us to explore more advanced mathematical concepts, such as convergence, limits, and determining the sum of infinitely many terms in some cases.

Alternating Series

An **alternating series** is a special type of series where the terms alternate between positive and negative. This happens often in sequences defined by a general term that involves an expression like \((-1)^{n+1}\).

In each term:

• - If n is even, \((-1)^{n+1}\) becomes negative, resulting in a negative term.
• - If n is odd, \((-1)^{n+1}\) becomes positive, giving a positive term.

Alternating series can converge to a limit even if their terms don't approach zero as quickly as in non-alternating series. This convergence is guided by the famous **Alternating Series Test**, which ensures convergence if:

• the absolute value of the terms decreases steadily, and
• the limit of the terms is zero as n approaches infinity.

Recognizing an alternating series can be challenging, but they often feature vital patterns that make their study fascinating and rewarding.

Summation in Mathematics

**Summation** is a mathematical operation that involves adding up a series of numbers or terms. The notation used for this operation is called **Sigma Notation** (\(\Sigma\) notation). The \(\Sigma\) symbol is a Greek letter that represents summation, and it is used to express long sums succinctly.

When using sigma notation:

• The symbol \(\sum\) is followed by an expression that defines the terms to be added.
• Below or beside \(\Sigma\), there's an index that usually starts with a specified initial value, continues with subsequent terms, and stops at the final term.
• The general form is \(\sum_{n=a}^{b} f(n)\), where \(a\) and \(b\) are the limits of summation and \(f(n)\) is the rule or formula for the individual terms.

This notation allows mathematicians to quickly and clearly represent the sum of many terms without writing each addition out. It shows up in a wide array of mathematical fields beyond series and sequences, from calculus to statistics, and beyond.