Question

Find the sum of the infinite geometric series, if it exists. $$ \sum_{k=1}^{\infty}-6\left(\frac{3}{2}\right)^{k-1} $$

Short answer

The given series does not have a sum (The sum is undefined) because the common ratio is not within the interval (-1,1).

Step-by-step solution

Determine convergence or divergence

The condition for a geometric series to converge is that the absolute value of the ratio, \(r\), is less than one, i.e. \(|-3/2| < 1\). Clearly, this series does not meet this condition, so it does not converge.

Provide answer based on convergence

Once we have determined a series does not converge, we can say that the sum of the series is undefined. No real number value can represent the sum.

Key concepts

Series Convergence

When we talk about series convergence, we're referring to whether the sum of an infinite series reaches a finite value as the number of terms grows indefinitely. For a geometric series, this happens when the common ratio between consecutive terms is between -1 and 1, exclusive. In the given exercise, the series is \(\sum_{k=1}^{\infty}-6\left(\frac{3}{2}\right)^{k-1}\), and the common ratio is \(\frac{3}{2}\). Since \(\left|\frac{3}{2}\right| > 1\), the series diverges, meaning that the sum is not finite as we add more and more terms.

Understanding convergence is crucial because it can affect how we interpret series in various contexts, such as calculating interests in finance or analyzing frequencies in signal processing. A converging series sanctions us to pursue a specific value, which can be crucial for precise calculations and predictions.

Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the 'common ratio'. For instance, in the series \(\sum_{k=1}^{\infty}-6\left(\frac{3}{2}\right)^{k-1}\), the first term is -6, and the common ratio is \(\frac{3}{2}\).

Each term gets successively larger (or smaller if the common ratio is between -1 and 0) by this ratio. Understanding geometric progressions is essential for deciphering growth patterns, such as those found in biology (like bacterial growth) or economics (like inflation rates). Mastery of this concept allows for the extrapolation and handling of data that follows exponential patterns.

Divergence

The concept of divergence comes into play when the sum of a series does not converge; in other words, it does not approach a specific value as more terms are added. Instead, the series either increases or decreases without bound or oscillates without settling to a point. This property is seen in our exercise, since the series’ common ratio exceeds the threshold for convergence.

Grasping the idea of divergence is important for recognizing patterns that are unsustainable or unbounded, such as overheating in a mechanical system, or in economic terms, hyperinflation. Divergent series often signal the need for reevaluation of a model or system to impose constraints that allow for stabilization.

Sum of a Series

The sum of a series is the cumulative result of adding all its terms together. For an infinite geometric series to have a sum, as we've learnt, it needs to be convergent. The sum of an infinite geometric series can be calculated using the formula \(\frac{a_1}{1 - r}\), where \(a_1\) is the first term and \(r\) is the common ratio. However, this particular exercise results in a divergent series, which means that the sum is not defined because the series grows without limit.

Knowing how to find the sum is useful in a variety of mathematical and practical applications, such as analyzing series circuits in physics or determining the present value of an annuity in finance. Recognizing that not all series can be summed is equally vital, saving time from attempting the impossible and focusing efforts on finding alternative approaches or solutions.