Problem 58 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1-1+1-1+\cdots $$

Step-by-Step Solution

Answer

The series is divergent as its common ratio \(r = -1\) does not meet the convergence criteria.

Step 1: Identify the Series

The series given is \(1 - 1 + 1 - 1 + \cdots\). This is an infinite geometric series with the first term \(a = 1\) and the common ratio \(r = -1\).

Step 2: Check for Convergence

A geometric series converges if the absolute value of the common ratio \(|r| < 1\). Here, the common ratio is \(r = -1\), and \(|-1| = 1\). Since \(|r| \geq 1\), this series does not satisfy the convergence condition.

Step 3: Conclusion - Divergence

Since the common ratio \(r = -1\) does not satisfy the condition \(|r| < 1\) for convergence, the series \(1 - 1 + 1 - 1 + \cdots\) is divergent.

Key Concepts

Convergence and Divergence

In the world of infinite series, determining whether a series converges or diverges is crucial. Simply put, convergence means that as you add more and more terms, the series approaches a specific finite value. Divergence, on the other hand, indicates that the series either keeps growing without bound or does not settle to a specific number.

To check for convergence in an infinite geometric series, one important criterion involves the common ratio (denoted as \( r \)). The rule is straightforward: If the absolute value of \( r \) is less than 1, \( |r| < 1 \), the series will converge. This means there's a sum to be found! If the absolute value of \( r \) is equal to or greater than 1, \( |r| \geq 1 \), the series diverges. No sum here because the series just won’t settle down!

In the given exercise, the series \(1 - 1 + 1 - 1 + \cdots\) has \( r = -1 \). Because \( |-1| = 1 \), we clearly see it does not fit the convergence rule, leading us to conclude that this series is divergent.

Geometric Series

A geometric series is a special type of series where each term after the first is found by multiplying the previous term by a constant called the common ratio (\( r \)). Don't be afraid of its name — it's just as simple as finding patterns in your favorite songs or dances!

The series usually takes the form:

• First term: \( a \)
• Second term: \( ar \)
• Third term: \( ar^2 \)
• ... and it continues like \( ar^3, ar^4, \ldots \)

In the infinite geometric series, you can have an endless number of terms. But whether it adds up to a specific, finite number depends on its convergence, which is dictated by the size of \( |r| \).

For instance, in our exercise, the series \( 1 - 1 + 1 - 1 + \cdots \) can be written as \( a, ar, ar^2, \ldots \) for \( a = 1 \) and \( r = -1 \). This means subsequent terms alternate between 1 and -1. For it to be a converging series, \( |r| \) must be less than 1, but as seen here, \( |r| = 1 \).

Common Ratio

The common ratio is a key ingredient in defining a geometric series. It's like the magical key that transforms one term of the series into the next by multiplication. In math terms, if you know the first term \( a \), and you have the common ratio \( r \), you can produce every other term in the series.

Mathematically, for a geometric series:

• First term: \( a \)
• Second term: \( ar \)
• \( n^{th} \) term: \( ar^{n-1} \)

Determining the common ratio involves dividing any term in the series by its preceding term. If the series has the form \( 1 - 1 + 1 - 1 + \, \ldots \), as in the exercise, the ratio is \(-1\). Each term is the negative of the one before.

Notice how essential \( r \) is for understanding the behavior of the series. The absolute value of \( r \) helps us know if the series converges like a peaceful monk reaching enlightenment or diverges like a tourist lost in a giant city.