Problem 59 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 3-3(1.1)+3(1.1)^{2}-3(1.1)^{3}+\cdots $$

Step-by-Step Solution

Answer

The series is divergent because \(|r| > 1\).

Step 1: Identify Common Ratio

The given series is an infinite geometric series in the form: \( a - ar + ar^2 - ar^3 + \ldots \)where \(a = 3\) and \(r = 1.1\). The common ratio \(r\) is the term by which we multiply to get from one term to the next.

Step 2: Check Convergence Condition

An infinite geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( |1.1| = 1.1 \), which is greater than 1. Therefore, the series does not meet the convergence condition.

Step 3: Determine Divergence

Since the condition for convergence is not met due to \( |r| > 1 \), this series is divergent. A divergent series does not have a finite sum.

Key Concepts

Convergence

In the context of infinite geometric series, convergence occurs when the series approaches a finite limit. This means as you sum more and more terms of the series, the total sum gets closer to a specific value rather than increasing indefinitely. For convergence to happen in a geometric series, the absolute value of the common ratio \( |r| \) must be less than 1.

When a geometric series converges, we can actually calculate its sum using the formula:
\[S = \frac{a}{1-r}\]
where \(a\) is the first term of the series and \(r\) is the common ratio.

• If \(|r| < 1\), the series converges to a finite number.
• Otherwise, the series diverges and doesn't have a finite sum.

Divergence

Divergence in an infinite geometric series happens when the series does not approach a finite limit. Instead, the sum continues to grow larger without bound. For an infinite series to diverge, the absolute value of its common ratio \( |r| \) needs to be 1 or greater.

In such cases, adding more and more terms only leads to a bigger sum, and it never settles on a single value. This explains why divergent series do not have a finite sum. In the given example with the common ratio \( r = 1.1 \), because \( |1.1| > 1 \), the series is clearly divergent. Hence, it's impossible to determine a single number that represents the sum of such a series.

• A divergent series will never settle on a specific finite sum.
• Divergence in geometric series indicates that the sequence escapes to infinity.

Common Ratio

The common ratio in a geometric series is a crucial element that determines how the series progresses from one term to the next. It is the factor by which we multiply each term to get to the subsequent term in the sequence. In mathematical form, if you have a series such as \( a, ar, ar^2, ar^3, \ldots \), the common ratio \( r \) is constant throughout the series.

The value of this ratio helps determine whether the series converges or diverges:

• For \(|r| < 1\), the terms decrease in magnitude, leading to convergence.
• For \(|r| \geq 1\), the terms grow or oscillate, resulting in divergence.

Identifying \( r \) early on is key to analyzing the behavior of the entire series.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is derived by multiplying the previous one by a fixed, non-zero number called the common ratio. In its simplest form, a geometric series looks like this: \( a, ar, ar^2, ar^3, \ldots \).

The infinite geometric series can either converge or diverge based on the value of the common ratio \( r \). It is particularly useful in various calculations and applications due to its predictable pattern of growth or decay.

• When convergent, the series sums up to a finite number.
• When divergent, the series fails to sum to a finite limit.

Understanding the nature of a geometric series is fundamental for tackling complex mathematical problems.