Question

Express each sum using summation notation. \(\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\cdots+(-1)^{12}\left(\frac{2}{3}\right)^{11}\)

Short answer

\( \sum_{n=1}^{12} (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \).

Step-by-step solution

Identify the general term in the series

Observe the pattern of the series. The first term is \(\frac{2}{3}\), the second term is \(-\frac{4}{9}\), the third term is \(\frac{8}{27}\), and so on. Notice that each term involves a power of \(\frac{2}{3}\) with alternating signs.

Formulate the general term

From the pattern, the general term for the \(n\)-th term can be written as \[a_n = (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1}\]. This represents the alternating signs and the increasing powers of \( \frac{2}{3} \).

Determine the range of the summation

The given series starts with \(n = 1\) and has the last term when \( n = 12 \). Therefore, the series runs from \(n = 1\) to \(n = 12\).

Combine into summation notation

Using the general term and range from the previous steps, the summation notation can be formulated as: \[\begin{aligned} \sum_{n=1}^{12} (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \end{aligned}\].

Key concepts

series

In mathematics, a series is the sum of the terms of a sequence.
Sequences are ordered lists of numbers, while series take things a step further by adding up the terms. For example, in the sequence 1, 2, 3, 4..., the series would be 1 + 2 + 3 + 4...
This concept is crucial in various domains of mathematics, including calculus and algebra. By understanding series, we can analyze infinite sums, determine convergence or divergence, and solve a range of mathematical problems.

general term

The general term of a series is a formula that allows you to find any term in the sequence by plugging in its position number. It is commonly denoted as \(a_n\) where \(n\) is the position in the sequence. For instance, in our exercise, each term involves a power of \( \frac{2}{3} \) alongside alternating signs. The pattern in the series allows us to formulate the general term as:
\[ a_n = (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \]
This formula is useful because it doesn't just tell us the term's value but also reveals the structure and behavior of the entire series. Identifying the general term is a key step in converting a list of terms into summation notation.

alternating series

An alternating series is a series where the signs of the terms alternate between positive and negative. This is usually indicated by a factor of \((-1)^n\) in the general term.
In our exercise, the series alternates, starting with a positive term followed by a negative term, and so on. This alternation can be seen clearly in the general term: \( (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \)
Understanding alternating series is important because they have unique convergence properties. Alternating series often converge even when their non-alternating counterparts do not, making them particularly interesting and useful in mathematical analysis.

summation formula

Summation notation is a shorthand way to represent the sum of a series. It uses the Greek letter sigma (\( \Sigma \)) along with an expression for the terms of the series. In summation notation for our series, it’s written as:
\[ \sum_{n=1}^{12} (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \]
This expression tells us to sum the terms from \(n = 1\) to \(n = 12\) where each term follows the pattern given by the general term \( (-1)^{n-1} \left( \frac{2}{3} \right)^{n-1} \)
Summation notation simplifies working with long series and makes the structure of the series clear at a glance. It's an essential tool in higher mathematics, enabling concise and efficient communication of complex sums.