Barisan dan Deret Bagian 5 - Deret Geometri Tak Hingga Matematika Wajib Kelas 11

Introduction

This tutorial provides a comprehensive overview of infinite geometric series as part of the mathematics curriculum for 11th grade. Understanding infinite geometric series is essential for solving various mathematical problems related to sequences and series. This guide will break down the concepts and calculations involved in infinite geometric series, making it easier for students to grasp.

Step 1: Understanding Geometric Series

• A geometric series is the sum of the terms of a geometric sequence.
• A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, called the common ratio (r).
• The general form of a geometric series is:
  • First term: a
  • Common ratio: r
  • The n-th term can be expressed as:
    • ( a_n = a \cdot r^{(n-1)} )

Step 2: Identifying Infinite Geometric Series

• An infinite geometric series is the sum of an infinite number of terms in a geometric sequence.
• It can be expressed as:
  • ( S = a + ar + ar^2 + ar^3 + \ldots )
• An infinite geometric series converges (has a finite sum) only if the absolute value of the common ratio is less than 1 (|r| < 1).

Step 3: Formula for the Sum of an Infinite Geometric Series

• If |r| < 1, the sum ( S ) of an infinite geometric series can be calculated using the formula:
  • ( S = \frac{a}{1 - r} )
• Here, ( a ) is the first term, and ( r ) is the common ratio.

Step 4: Example Calculation

1. Identify the first term and common ratio:
   • Example: First term, ( a = 5 ); common ratio, ( r = \frac{1}{2} )
2. Check the condition for convergence:
   • Since |r| = 0.5 < 1, the series converges.
3. Apply the formula:
   • Calculate the sum:
     • ( S = \frac{5}{1 - \frac{1}{2}} = \frac{5}{0.5} = 10 )

Step 5: Practical Applications

• Infinite geometric series can be applied in various fields such as finance (calculating loan payments), physics (analyzing wave patterns), and computer science (algorithm analysis).
• Understanding these concepts will help in solving real-world problems involving rates of growth and decay.

Conclusion

In this tutorial, we explored the concept of infinite geometric series, identified their properties, and learned how to calculate their sums using a specific formula. To further enhance your understanding, practice by solving additional problems involving different values of a and r, ensuring you check the convergence condition. For more advanced topics, consider exploring series tests and their applications in calculus.