Question

Write the following series in the abbreviated \(\sum\) form. $$ \frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+\cdots $$

Short answer

\( \sum_{n=2}^{\text{to infinity}} (-1)^{n+1}\frac{1}{n^2} \)

Step-by-step solution

Identify the General Term

Examine the given series to recognize a pattern in the terms. The terms alternate in sign and the denominators are the squares of consecutive integers starting from 2.So, the general term can be written as: decreasing: \(-1\)pos. term: \(\frac{1}{n^2}\) where \(n\) starts from 2.Therefore, the general term is: \((-1)^{n+1}\frac{1}{n^2}\) starting from \(n=2\).

Write the Series in Summation Form

Now that the general term is identified, express the series using the summation notation \( \sum \). The series starts at \(n = 2\), and alternates signs with each term:\[ \sum_{n=2}^{\text{to infinity}} (-1)^{n+1}\frac{1}{n^2} \]

Key concepts

Series Representation

In mathematics, a series is the sum of the terms of a sequence. Representing a series in summation notation, \(\sum\), helps to write it concisely. Instead of listing every term, summation notation provides a compact form. For example, the series \(\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+\text{…}\) is written using \(\sum\) to show the repeating pattern.Understanding summation notation is essential. \(\sum_{n=1}^{\infty} a_n\) represents the series whose terms are given by \a_n\, starting from \=1\ to infinity. This helps in handling infinite series efficiently and understanding their behavior.

General Term Identification

Identifying the general term of a series is critical to its representation. A general term describes each term in a sequence using a simple formula. For instance, in the series \(\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+\text{…}\), we notice the following:

• The denominators are squares of consecutive integers starting from 2: \4=2^2,\text{ }9=3^2, \text{ }16=4^2,\text{ }25=5^2 \text{…}\
• The signs alternate between positive and negative.

So, the general term can be expressed using \(n\) as \(\frac{1}{n^2}\), where \(n\) starts at 2. However, since the terms alternate in sign, we introduce \( (-1)^{n+1}\), making the general term \((-1)^{n+1}\frac{1}{n^2}\). This general term helps in expressing the entire series compactly.

Alternating Series

An alternating series is a series where the terms change signs alternatively, typically from positive to negative. The series \(\frac{1}{4} - \frac{1}{9} + \frac{1}{16} - \frac{1}{25} + \text{…}\) is an example. This kind of series can be written using \( (-1)^n\) or similar variations to represent the sign change. Key points about alternating series:

• Each term’s sign is different from the one before it.
• They can converge to a specific value, depending on the terms.

The given series is rephrased in summation notation as: \[ \sum_{n=2}^{\infty} (-1)^{n+1}\frac{1}{n^2} \], capturing the alternating signs and squared terms succinctly. Understanding alternating series is vital in analyzing series convergence and behavior.