Problem 32 - Single Variable Calculus

Question

Determine whether the given series converges or diverges. If it converges, find its sum. \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{5^{n}}\)

Step-by-Step Solution

Answer

The given series is a convergent geometric series with first term \(a = 3\) and common ratio \(r = \frac{3}{5}\). By using the formula \(S = \frac{a}{1-r}\), the sum of the series is found to be 15.

Step 1: Rewrite the Series to Identify \(a\) and \(r\)

Express the series as \(\sum_{n=0}^{\infty} 3 \cdot \left(\frac{3}{5}\right)^n\). From this representation, it's clear that the first time \(a\) is 3 and the common ratio \(r\) is \(\frac{3}{5}\).

Step 2: Determine Convergence or Divergence

To establish whether the series is convergent or divergent, compare the common ratio \(r\) with 1. If the absolute value of \(r\) is less than 1, the series is convergent. If it's greater than or equal to 1, it is divergent. By evaluation, \(|r| = \left|\frac{3}{5}\right| = 0.6\), which is less than 1. Therefore, the series converges.

Step 3: Calculate the Sum of the Series

Now that it is established that the series is convergent, the sum of the series can be calculated using the formula \(S = \frac{a}{1-r}\). Substitute \(a = 3\) and \(r = \frac{3}{5}\) into the formula: \[S = \frac{3}{1-\frac{3}{5}} = 15\] The sum of the series is 15.

Key Concepts

Understanding Series Convergence

Let's begin by understanding how we decide if a series converges or diverges. A series is like a sequence of numbers added together infinitely. To see if adding them keeps growing endlessly (diverges) or settles down to a fixed number (converges), we look at the common ratio, denoted as \( r \). If the series is a geometric one, like in this exercise, it has a pattern of multiplying by \( r \) as we move along the series.

• If the absolute value of \( r \) is less than 1, the series converges, meaning it will approach a specific value as you add more terms.
• If the absolute value of \( r \) is equal to or greater than 1, the series does not settle to a number and is said to diverge.

This simple rule makes it easy to determine convergence, which is what we identify first before moving forward with our calculations.

The Role of the Common Ratio

In a geometric series, the common ratio \( r \) plays a crucial role. It is the factor by which we multiply each term to get to the next term in the series. Think of each term as an incrementally scaled version of the one before. In our given series, we had to rewrite it to clearly see the pattern and identify \( r \), which was \( \frac{3}{5} \). This value is important because:

• It tells us how quickly the terms are growing or shrinking compared to each other.
• It helps us apply the convergence rule in determining the behavior of the series.

When \( |r| < 1 \), like \( 0.6 \) in this specific case, it confirms convergence, which sets the stage for further calculations.

Calculating the Sum of a Convergent Series

Once the series is confirmed to converge, we can find its sum. This is one of the advantages of geometric series because we have a neat formula to do so: \[ S = \frac{a}{1-r} \] where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio from earlier. Applying it to our series, where \( a = 3 \) and \( r = 0.6 \), we substitute into the formula to get: \[ S = \frac{3}{1-0.6} = 15 \] So, the series doesn’t just end up somewhere, but it perfectly adds up to 15. This calculation shows the power of geometric series in simplifying complex, infinite sums to easily manageable numbers.