Problem 21 - Essential Calculus

Question

Verify that the infinite series converges. $$ \sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^{n} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^{n}\) is a convergent series.

Step 1: Identify the geometric series and its ratio

The series \(\sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^{n}\) is a geometric series, where the constant factor between terms (the ratio) is \(r = \frac{3}{4}\). This is calculated by dividing one term of the series by the previous term.

Step 2: Verification of the ratio

A geometric series will converge if the absolute value of the ratio r is less than 1. So, we check the magnitude of r: \(|r| = |\frac{3}{4}| < 1\). Since the condition is met, the series converges.

Step 3: Conclusion

The series \(\sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^{n}\) is deemed a convergent series, because the ratio is such that its absolute is less than 1.

Key Concepts

Geometric Series

A geometric series is a series where every term after the first is found by multiplying the previous one by a fixed, non-zero number called the "ratio." In mathematical terms, a geometric series has the form \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio. This type of series is fundamental in mathematics due to its predictable pattern and wide applications.

• The terms grow or shrink exponentially based on the ratio.
• If the ratio \( |r| < 1 \), the series is said to converge, which means it approaches a specific finite sum.
• Conversely, if \( |r| \geq 1 \), the series diverges, growing without bound.

Understanding geometric series helps us to analyze growth processes in science and finance, such as calculating compound interest or the decay of materials. The convergence criterion is straightforward, making it a useful tool for evaluating infinite series.

Ratio Test

The ratio test is a mathematical technique to determine the convergence or divergence of an infinite series. This test is particularly useful because it can be applied to a wide range of series. To use the ratio test:

• Consider the series \( \sum a_n \) with terms \( a_n \).
• Calculate \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• Interpret the value of \( L \):
  • If \( L < 1 \), the series converges absolutely.
  • If \( L > 1 \) or \( L \) is infinite, the series diverges.
  • If \( L = 1 \), the test is inconclusive, and another method is required.

In the context of geometric series, the ratio test seamlessly complements their analysis, reaffirming convergence when \( |r| < 1 \). By calculating \( L \), the test provides a straightforward approach for recognizing series behavior, making it a vital tool for students and professionals.

Convergent Series

A convergent series is one in which the sum of its terms approaches a specific finite number as more terms are added. This property is essential because it indicates that an infinite sum can have a meaningful total. Specifically, convergence means:

• There exists a limit \( S = \lim_{N \to \infty} \sum_{n=1}^{N} a_n \) for the series \( \sum a_n \).
• This implies that as we progressively add more terms, the partial sums approach \( S \).

For geometric series, convergence is best understood through the common ratio. If \( |r| < 1 \), then the series will converge to the sum given by the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term. This succinct formula provides a straightforward way to find the finite sum of an infinite sequence, which is immensely useful in mathematical calculations and practical applications.
Convergent series are foundational in calculus and have applications ranging from physics to economics, where they model real-world phenomena that require summing infinite sequences.