Question

Determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} n \cdot\left(\frac{3}{5}\right)^{n}$$

Short answer

Since \( L = \frac{3}{5} < 1 \), the series converges by the Ratio Test.

Step-by-step solution

Identify the Series Terms

The given series is \( \sum_{n=1}^{\infty} n \left( \frac{3}{5} \right)^n \). Each term of the series can be expressed as \( a_n = n \left( \frac{3}{5} \right)^n \).

Apply the Ratio Test

The Ratio Test involves checking the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). First compute \( a_{n+1} = (n+1) \left( \frac{3}{5} \right)^{n+1} \).

Simplify the Ratio

Calculate \( \frac{a_{n+1}}{a_n} = \frac{(n+1)\left(\frac{3}{5}\right)^{n+1}}{n\left(\frac{3}{5}\right)^n} = \frac{(n+1)}{n} \cdot \left(\frac{3}{5}\right) \).

Evaluate the Limit

Now find \( L = \lim_{n \to \infty} \frac{(n+1)}{n} \cdot \left(\frac{3}{5}\right) = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) \cdot \left(\frac{3}{5}\right) = 1 \cdot \frac{3}{5} = \frac{3}{5} \).

Determine Convergence from Ratio Test

Since \( L = \frac{3}{5} < 1 \), the Ratio Test concludes that the series \( \sum_{n=1}^{\infty} n \left( \frac{3}{5} \right)^n \) converges.

Key concepts

Convergence of Series

In the world of infinite series, "convergence" is a key concept that tells us whether the sum of an infinite list of numbers results in a finite value. When we talk about a series converging, we mean that as you keep adding more and more terms, the total sum approaches a particular finite number. This is very different from divergence, where the sum just keeps growing without heading towards any specific limit.

For a series to converge, a specific test is often required to determine this property. One popular method is the Ratio Test, which provides a simple mechanism to check convergence. If you apply the Ratio Test and find that the result is less than 1, the series converges. In our example, the computed ratio was \(\frac{3}{5}\), clearly less than 1, indicating convergence.

Understanding whether a series converges is important across various fields of science and engineering. This is because it helps in estimating, approximating, and controlling solutions in real-world applications.

Infinite Series

When you hear about "infinite series," think of it as adding up an endless number of terms. Each term in the series gets added to the previous sum to form a 'partial sum', but because there are infinitely many terms, you never actually finish the list.

An infinite series can be written in a compact form using the sigma notation \(\sum_{n=1}^{\infty} a_n\), which simply means you're adding up terms \(a_1, a_2, a_3,...\) to infinity. In our exercise, the infinite series is given by \(\sum_{n=1}^{\infty} n \left(\frac{3}{5}\right)^n\), showing an endless sum of the terms specific to the formula \(n \left(\frac{3}{5}\right)^n\).

However, not all infinite series have a meaningful sum or approach a particular number. That's why various tests, like the Ratio Test, are essential tools to determine whether an infinite series converges (has a sensible sum) or diverges (just keeps growing or oscillates indefinitely).

Calculus

Calculus is a branch of mathematics that provides tools for analyzing change and motion. It largely comprises two main ideas: derivatives (dealing with rate of change) and integrals (focusing on accumulation). Infinite series and the concept of convergence play a vital role in calculus, as they help in evaluating functions, solving limits, and creating precise mathematical models of real-world scenarios.

For calculus students, understanding series convergence is essential because it aids in evaluating more complex mathematical concepts, such as differentiating and integrating power series. The Ratio Test is one of many methods used in calculus to determine convergence, providing a clear mechanism to tackle infinite sums.

The ability to handle infinite series with ease allows mathematicians, scientists, and engineers to simplify and solve problems that involve summing infinite data points. In our exercise, the series \(\sum_{n=1}^{\infty} n \left(\frac{3}{5}\right)^n\) was tested for convergence using calculus-based techniques, demonstrating how calculus applies to infinite series in practical scenarios.