Question

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 6\left(\frac{4}{5}\right)^{n} $$

Short answer

The sum of the given convergent series is 30.

Step-by-step solution

Understanding the Geometric Series

A geometric series is a series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. The infinite geometric series takes the form \(S = \frac{a}{1-r}\), where 'a' is the first term, and 'r' is the ratio. In the given series, \(a = 6\) and \(r = \frac{4}{5}\).

Using the Infinite Geometric Series Formula

Plugging 'a' and 'r' into the formula to find the sum of the series: \(S = \frac{6}{1-\frac{4}{5}}\).

Calculation

Simplify the expression to find the sum. After performing the calculations, you get: \(S = 30\).

Key concepts

Infinite Geometric Series

An infinite geometric series is a fascinating concept in mathematics where you sum an endless sequence of terms, each one being a fixed multiple (or ratio) of the one before it. To put it simply, you start with a number and keep multiplying it by the same value, creating a series that goes on forever.

The beauty of this series is that even though it has infinite terms, it can often converge to a finite value, but only if the absolute value of the ratio is less than one! Why is that? Well, because as you keep multiplying by a fraction (a number between -1 and 1), the terms get smaller and smaller, eventually approaching zero. This shrinking effect allows the sum of all those terms to reach a stable, finite number.

However, not all infinite series behave this way. If the ratio is 1 or greater in absolute value, the terms won't shrink, and the series diverges, meaning it doesn't sum up to a specific number but rather grows infinitely.

Series Sum Calculation

The series sum calculation for an infinite geometric series might seem like magic at first glance. But it's all based on a clever formula that allows us to determine the sum quickly. This formula, known as the sum formula for an infinite geometric series, is given by \( S = \frac{a}{1 - r} \), where 'S' is the sum, 'a' is the first term of the series, and 'r' is the common ratio.

To use this formula, you should initially verify that the series is indeed convergent (which means \( |r| < 1 \)). Once you've established convergence, you can confidently plug in the values into the formula. The simplicity of this formula masks the complex calculus concepts underlying the summation of infinitely many terms but remains a powerful tool in understanding and computing convergent series. Mathematically, it encapsulates the idea that the sum of an infinite series is equal to the first term divided by the difference between one and the ratio.

Geometric Series Ratio

At the heart of any geometric series lies the ratio, often denoted as 'r'. The ratio is what we multiply each term by to get to the next one. It's the soul of the geometric progression, determining the behavior of the series — whether it will converge or diverge.

In the case of an infinite series, the ratio's value is critical because it decides if we can even find a finite sum. For convergence, the ratio must be strictly between -1 and 1, not inclusive. If 'r' is greater or equal to 1, or less than or equal to -1, then each term will either stay the same or grow larger, leading to a divergent series. In your homework problem, the ratio \( r = \frac{4}{5} \), which is less than 1, meaning the series indeed converges, and we can use the formula to find its sum.