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. $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ 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: $$\sum _{n=2}^{\mathrm{\infty }}(-1{)}^{n+1}\frac{1}{n^2}$$, capturing the alternating signs and squared terms succinctly. Understanding alternating series is vital in analyzing series convergence and behavior.