Problem 39 - Contemporary Precalculus

Question

Express the sum in \(\Sigma\) notation. $$2+\frac{2^{2}}{2}+\frac{2^{3}}{3}+\frac{2^{4}}{4}+\frac{2^{5}}{5}+\frac{2^{6}}{6}+\frac{2^{7}}{7}$$

Step-by-Step Solution

Answer

Question: Express the given sum in Sigma notation: $\frac{2^1}{1} + \frac{2^2}{2} + \frac{2^3}{3} + \frac{2^4}{4} + \frac{2^5}{5} + \frac{2^6}{6} + \frac{2^7}{7}$. Answer: $\sum_{n=1}^{7} \frac{2^{n}}{n}$

Step 1: Identifying the pattern

From the given sum, we can see that each term is a power of 2 divided by an integer. The exponent of 2 starts from 1 and goes up to 7. The denominators of the fractions also start from 1 and go up to 7.

Step 2: Formulating the general term

Now that we have identified the pattern, we can write a general term for the sum. The general term for this sum can be expressed as $$\frac{2^{n}}{n}$$ where n starts at 1 and goes up to 7.

Step 3: Expressing the sum in Sigma notation

Based on the general term we derived in Step 2, we can now express the sum in Sigma notation. The sum can be written as: $$\sum_{n=1}^{7} \frac{2^{n}}{n}$$

Key Concepts

Sequences and Series

When we talk about sequences in mathematics, we refer to an ordered list of numbers that follow a specific pattern or rule. Each number in a sequence is called a term. Sequences can be finite or infinite, depending on whether they have a last term or not.

Now, a series is essentially the sum of the terms of a sequence. When we write down the series explicitly, like in the exercise 2 + \(\frac{2^2}{2}\) + \(\frac{2^3}{3}\) + \(\frac{2^4}{4}\) + \(\frac{2^5}{5}\) + \(\frac{2^6}{6}\) + \(\frac{2^7}{7}\), we are adding each term of a sequence together. If we were to extend this series, following the pattern, we would be delving into the realm of infinite series.

In the context of the given exercise, sigma notation is an efficient way to represent the sum of a series. It allows us to succinctly express long sums like the one we see above, thus providing a clearer and more compact representation of the series.

Exponential Expressions

Exponential expressions are a type of mathematical notation where a number (the base) is raised to the power of an exponent. The exponent signifies how many times the base is multiplied by itself. In simple terms, if we have an expression like \(2^3\), this means 2 is multiplied by itself 3 times, resulting in 8.

In the exercise provided, the series includes terms with increasing powers of 2. Understanding how to manipulate exponential expressions is crucial in simplifying and working with series that include exponential terms. When we write the solution to this exercise in sigma notation, we are summing up each of these exponential expressions, effectively condensing a potentially lengthy series into a concise formula.

General Term of a Sequence

The general term of a sequence is an algebraic expression that defines how to find any term in the sequence. In the exercise, the part of the solution where it is stated, the general term for this sum can be expressed as \(\frac{2^n}{n}\), encapsulates the pattern of each term in the series. 'n' represents the position of the term in the sequence, and as 'n' changes, we can calculate different terms.

By identifying this general term, students are able to determine any term in the sequence without having to list out all the preceding terms. This becomes particularly handy for very long sequences or when we need to understand the properties of the series it sums to. Using the general term \(\frac{2^n}{n}\), we can also conveniently express the series using sigma notation, as was demonstrated in the solution \(\sum_{n=1}^{7} \frac{2^{n}}{n}\), streamlining the original complex expression.