Problem 70 - Essential Calculus

Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{n}{500}\right)^{n} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Write down the nth term of the series

The nth term of the series is \(\left(\frac{n}{500}\right)^{n}\)

Step 2: Apply the Root Test

The Root Test involves taking the nth root of the absolute value of the nth term. In this case, because the nth term is always positive, we don't need to take the absolute value. The sequence becomes \(\sqrt[n]{\left(\frac{n}{500}\right)^{n}}\), which simplifies to \(\frac{n}{500}\)

Step 3: Evaluate the limit

As n approaches infinity, the sequence \(\frac{n}{500}\) also approaches infinity.

Step 4: Interpret the result

According to the Root Test, because the limit is greater than 1, the series \(\sum_{n=1}^{\infty}\left(\frac{n}{500}\right)^{n}\) diverges.

Key Concepts

Infinite Series

An infinite series is basically a sum of an infinite sequence of terms. It’s like adding up an endless list of numbers, which may sound impractical, but in mathematics, such series can often be summed up to yield a finite value, or they might grow without bound, approaching infinity. The series in our exercise,
\[ \sum_{n=1}^{\infty}\left(\frac{n}{500}\right)^{n} \]
is an example of an infinite series; it's a sum that theoretically continues without end.

Understanding whether such a series converges to a finite value or diverges is crucial in many areas of mathematics and applied sciences. It impacts how we solve problems in physics, engineering, and economics, to name a few. Series that converge have a finite limit, they approach a specific number as you add up more and more terms. In contrast, series that diverge keep growing past any bound as you sum more terms.

Convergence and Divergence

The concepts of convergence and divergence are at the heart of understanding infinite series. When mathematicians talk about a series converging, they mean that as you add more terms, the total sum approaches a certain value. This is particularly interesting because even though you're adding infinitely many terms, they can add up to a perfectly finite number.

On the other hand, a series diverges if the sum does not approach any particular value as you add more terms. Instead of honing in on a specific number, the sum could grow infinitely large, oscillate between values, or behave erratically. In the case of the given exercise, the series diverges because the terms get larger as ‘n’ increases, which means the overall sum can never settle at a specific number.

Limits

The concept of a limit in mathematics refers to the value that a function or sequence 'approaches' as the index or input 'approaches' some value. Limits are fundamental to calculus and are used to define concepts like continuity, derivatives, and integrals.

To illustrate, let's look at the exercise where we evaluate the limit of the sequence
\( \lim_{n \to \infty} \frac{n}{500} \).
As 'n' grows larger and larger, this sequence doesn’t approach a fixed number. Instead, it keeps increasing, which indicates that there is no finite limit. This is a clear signal that our series does not have a sum that stabilizes at a particular value, hence it diverges. Understanding limits is crucial because it dictates the behavior of sequences and series and determines convergence or divergence.