Question

Using sigma notation, write the following expressions as infinite series. $$ 1-1+1-1+\cdots $$

Short answer

The infinite series is \(\sum_{n=1}^{\infty} (-1)^{n+1}\).

Step-by-step solution

Understand the alternating sequence

The sequence given is repeated with terms alternating between 1 and -1: \(1 - 1 + 1 - 1 + \cdots\). We need to represent this infinite sequence using sigma notation.

Identify the general term pattern

Notice that the odd terms (1st, 3rd, 5th, etc.) are 1, while the even terms (2nd, 4th, 6th, etc.) are -1. This can be described using the general term \((-1)^{n+1}\), where \(n\) stands for the term's position in the sequence.

Construct the sigma notation expression

Using the identified general term, setup the infinite series in sigma notation. The sum can be represented as: \[\sum_{n=1}^{\infty} (-1)^{n+1}\]This sums terms consecutively using \((-1)^{n+1}\) starting from \(n = 1\) to infinity.

Key concepts

Sigma Notation

Sigma notation is a powerful tool used to represent the sum of a sequence of terms. It is denoted by the Greek letter \(\Sigma\), which stands for "sum". This notation allows you to write a long sequence of additions in a concise way.

In sigma notation, the general format is:

• \(\sum_{n=a}^{b} f(n)\)

Here, \(n\) is the index of summation, \(a\) is the starting value, and \(b\) is the ending value of \(n\). The expression \(f(n)\) defines the general term of the sequence.

When you see an infinite series, like our example, you write \(\sum_{n=1}^{\infty} \) to represent a series that continues indefinitely. Using this notation keeps your expressions tidy, effectively communicating complex sums, which is especially handy in calculus when dealing with infinite sequences.

Alternating Sequence

An alternating sequence is a type of sequence where the signs of the terms alternate between positive and negative. In our exercise, the sequence is:

• \(1, -1, 1, -1, \ldots\)

Each term in the sequence switches its sign compared to the previous one.

This kind of sequence can generally be written using the expression \((-1)^{n}\), or in some cases \((-1)^{n+1}\), depending on whether you start with a positive or negative term. In our case, since we start with 1 and want it to appear in odd positions, we utilize \((-1)^{n+1}\). This expression alternates the sign because:

• When \(n\) is odd, \((-1)^{n+1}\) gives a positive term (1).
• When \(n\) is even, \((-1)^{n+1}\) results in a negative term (-1).

Understanding alternating sequence patterns is key when expressing infinite series in sigma notation.

General Term of a Sequence

The general term of a sequence is a formula that allows us to find any term in the sequence based on its position.

• It is expressed as \(f(n)\), where \(n\) represents the position of the term in the sequence.

In the given exercise, the sequence alternates between 1 and -1. Recognizing this pattern led us to the general term \((-1)^{n+1}\).

To find the general term:

• Observe the sequence's pattern or rule.
• Identify how each term relates to its position \(n\).

For our sequence:

• The terms are 1 when \(n\) is odd, and -1 when \(n\) is even. This confirms that \((-1)^{n+1}\) suits the pattern.

Once you've derived the general term, you can apply it within sigma notation to algebraically express the entire sequence.

Infinite Sum

An infinite sum, or infinite series, is a sum that extends indefinitely, meaning it has an infinite number of terms. For an infinite series,

• We typically use the notation \(\sum_{n=1}^{\infty}\), where \(n\) begins at some finite integer and extends to infinity.

In the exercise, the series \(1 - 1 + 1 - 1 + \cdots\) can be expressed with an infinite sum, as follows:

• \[\sum_{n=1}^{\infty} (-1)^{n+1}\]

This expression doesn't have an end, making it "infinite."

While not all infinite sums converge (i.e., approach a specific value), understanding their structure is crucial in calculus and advanced mathematics. Recognizing how to write and interpret these sums can open the door to solving sophisticated mathematical problems like finding limits.