### Working with Geometric Series: AP Calculus BC Study Guide

#### Introduction

Welcome to the world of geometric series! Let’s embark on this mathematical adventure where sequences meet series and convergence is king. If you've ever stacked dominoes and wondered if there’s a way to stack them infinitely without them toppling over—a geometric series might just be your answer. But remember, this is an **AP Calculus BC** only topic; if you’re in Calculus AB, skip ahead or marvel at the mathematical beauty from afar. 📚🚀

#### What is a Geometric Series?

A geometric series is not just any collection of numbers; it’s a well-behaved series that follows a specific format. Picture the series on a reality show runway, poised and perfect. Each term in this series is multiplied by a constant ratio to get the next term. Mathematically, our stars look like this:

For a series starting at 0: [ s_n = \sum_{n=0}^{\infty} a \cdot r^n ]

For a series that starts at 1: [ s_n = \sum_{n=1}^{\infty} a \cdot r^{n-1} ]

Here, (a) is the initial term, and (r) is the rocking ratio that keeps the series in check. No matter where you start your sum (0 or 1), the fundamental values of (a) and (r) remain unchanged, much like your love for pizza despite the toppings. 🍕

#### Building a Geometric Series from a Given Sequence

Creating a geometric series is like assembling the perfect playlist or outfit—order and consistency matter. Let’s jazz things up with a couple of examples.

##### Example 1:

Consider the sequence: [ 27, 9, 3, 1, \frac{1}{3}, \frac{1}{9},... ]

**Step 1**: Determine (a) and (r). Our initial term (a) is 27. To find (r), simply observe the ratio between any two consecutive terms. Using 27 and 9:

[ r = \frac{9}{27} = \frac{1}{3} ]

**Step 2**: Plug these values into your geometric series format:

For (n = 0): [ s_n = \sum_{n=0}^{\infty} 27 \cdot \left(\frac{1}{3}\right)^n ]

For (n = 1): [ s_n = \sum_{n=1}^{\infty} 27 \cdot \left(\frac{1}{3}\right)^{n-1} ]

Congratulations, you've nailed it! 🎉 Let’s try another.

##### Example 2:

Take the sequence: [ 2, -6, 18, -54,... ]

**Step 1**: Determine (a) and (r). Here, (a = 2). To find (r), consider:

[ r = \frac{-6}{2} = -3 ]

**Step 2**: Plug these values into your geometric series format:

For (n = 0): [ s_n = \sum_{n=0}^{\infty} 2 \cdot (-3)^n ]

For (n = 1): [ s_n = \sum_{n=1}^{\infty} 2 \cdot (-3)^{n-1} ]

Even with a negative ratio, you're doing great! Note that the absolute value of (r) will guide our convergence journey.

#### The Geometric Series Test

The Geometric Series Test is our magical theorem to determine whether our series will converge (chill with a finite sum) or diverge (run off to infinity).

A geometric series converges if: [ 0 < |r| < 1 ]

And diverges if: [ |r| \geq 1 ]

When a series converges, to find that elusive sum ( S ), use this nifty equation: [ \sum_{n=0}^{\infty} a \cdot r^n = \frac{a}{1-r} ]

Now, let’s test our examples.

##### Geometric Test: Example 1 (continued)

[ s_n = \sum_{n=0}^{\infty} 27 \cdot \left(\frac{1}{3}\right)^n ]

Here, ( r = \frac{1}{3} ). Since ( 0 < \frac{1}{3} < 1 ), the series converges. Using our sum formula:

[ \frac{27}{1 - \frac{1}{3}} = \frac{27}{\frac{2}{3}} = 40.5 ]

Congratulations! The sum of the series is 40.5, all thanks to the mighty geometric series test. 🎩✨

##### Geometric Test: Example 2 (continued)

[ s_n = \sum_{n=0}^{\infty} 2 \cdot (-3)^n ]

Here, ( r = -3 ). Since ( |-3| = 3 \geq 1 ), the series diverges. Cue dramatic no-no music: 🚨

#### Conclusion

Bravo! You've mastered your first convergence test. Whether series converge or diverge, remember: every sequence has its moment in the sun. Keep practicing, and soon you'll be the rock star of infinite geometric series. 🤓🌟

Now go forth, and let your newfound knowledge shine brighter than a supernova!