Problem 16
Question
Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ -\frac{1}{5}+\frac{2}{5}-\frac{3}{5}+\frac{4}{5}-\frac{5}{5} $$
Step-by-Step Solution
Verified Answer
Use \(\sum_{n=1}^{5} (-1)^n \cdot \frac{n}{5}\).
1Step 1: Identify the Pattern
Observe the sequence: \(-\frac{1}{5}, \frac{2}{5}, -\frac{3}{5}, \frac{4}{5}, -\frac{5}{5}\). Notice that the numerators are changing incrementally and the signs alternate between negative and positive.
2Step 2: Determine the General Term
Identify how each term follows a specific pattern. We see that the numerators are integers increasing by 1 each time. Also, notice the odd terms are negative, while the even terms are positive.
3Step 3: Define the Expression with Alternating Signs
Express the alternating signs using \((-1)^n\), where \(n\) starts at 1. This alternates signs starting with negative when \(n = 1\).
4Step 4: Formulate the General Term
The term can be expressed as \((-1)^n \cdot \frac{n}{5}\). This includes both the terms' alternating signs and their linear progression in the sequence.
5Step 5: Express in Sigma Notation
Combine the above results to express the sequence in sigma notation. The sum can be written as: \(\sum_{n=1}^{5} (-1)^n \cdot \frac{n}{5}\).
Key Concepts
Alternating SeriesSequence PatternGeneral Term Formula
Alternating Series
An alternating series is a sequence where the signs of its terms switch back and forth between positive and negative. This feature distinguishes it from other sequences and is a powerful tool in mathematical expressions.
To identify an alternating series, look for a consistent pattern in the sign changes. These patterns are often mathematically represented using \((-1)^n\), where \(-1\) indicates the alternation (positive or negative), and \(n\) determines the current position in the sequence. The series in the exercise, \(-\frac{1}{5} + \frac{2}{5} - \frac{3}{5} + \frac{4}{5} - \frac{5}{5}\), follows this concept.
If a series starts with a negative term, as in this case, \((-1)^n\) is used to maintain the switching pattern, with negative at odd positions and positive at even ones. You can generalize this idea to any series with alternating signs, making it easier to express complex sequences in a simplified form.
To identify an alternating series, look for a consistent pattern in the sign changes. These patterns are often mathematically represented using \((-1)^n\), where \(-1\) indicates the alternation (positive or negative), and \(n\) determines the current position in the sequence. The series in the exercise, \(-\frac{1}{5} + \frac{2}{5} - \frac{3}{5} + \frac{4}{5} - \frac{5}{5}\), follows this concept.
If a series starts with a negative term, as in this case, \((-1)^n\) is used to maintain the switching pattern, with negative at odd positions and positive at even ones. You can generalize this idea to any series with alternating signs, making it easier to express complex sequences in a simplified form.
Sequence Pattern
Understanding the pattern in a sequence is crucial for expressing it in a mathematical formula or notation. In the given sequence \(-\frac{1}{5}, \frac{2}{5}, -\frac{3}{5}, \frac{4}{5}, -\frac{5}{5}\), there's a pattern not just in alternating signs, but also in its numerators.
The numerators increment by one in each subsequent term. This signifies a linear increase characteristic of simple arithmetic progressions. Observing a sequence pattern involves identifying increments, decrements, or any repeated logical instructions that guide the entire series.
Recognizing patterns also allows you to predict future terms efficiently and to convert the sequence into a general term formula, which is the next logical step after understanding the underlying pattern.
The numerators increment by one in each subsequent term. This signifies a linear increase characteristic of simple arithmetic progressions. Observing a sequence pattern involves identifying increments, decrements, or any repeated logical instructions that guide the entire series.
Recognizing patterns also allows you to predict future terms efficiently and to convert the sequence into a general term formula, which is the next logical step after understanding the underlying pattern.
General Term Formula
The general term formula is a mathematical expression that defines any term in a sequence from its position number. It saves time and effort by providing a straightforward calculation method for any term within the series.
The exercise uses such a formula: \((-1)^n \cdot \frac{n}{5}\). This includes the alternating sign feature and a consistent denominator, with a numerator that corresponds directly to the position number \(n\).
Using this formula, you can express any term in the sequence without listing all previous terms, which is especially helpful for long sequences. If you want to find, say, the 10th term of a similar sequence, just plug \(10\) into the formula, and it will yield the desired result. Such formulas are invaluable in mathematical proofs, series evaluations, and simplifying sums using sigma notation.
The exercise uses such a formula: \((-1)^n \cdot \frac{n}{5}\). This includes the alternating sign feature and a consistent denominator, with a numerator that corresponds directly to the position number \(n\).
Using this formula, you can express any term in the sequence without listing all previous terms, which is especially helpful for long sequences. If you want to find, say, the 10th term of a similar sequence, just plug \(10\) into the formula, and it will yield the desired result. Such formulas are invaluable in mathematical proofs, series evaluations, and simplifying sums using sigma notation.
Other exercises in this chapter
Problem 15
In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-2}^{4}\left(\frac{x}{2}+3\right) d x $$
View solution Problem 15
In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of eq
View solution Problem 16
Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\pi / 3}^{\pi / 3} \frac{1-\cos 2 t}{2} d t $$
View solution Problem 16
In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{1 / 2}^{3 / 2}(-2 x+4) d x $$
View solution