Problem 29 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) $$

Step-by-Step Solution

Answer

The series diverges by the Limit Comparison Test with the harmonic series.

Step 1: Rewrite the Series

First, let's rewrite the series to see if there is any simplification possible. The given series is \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) \). This can be rewritten as \( \sum_{n=1}^{\infty} a_n \) where \( a_n = \frac{1}{n} - \frac{1}{n^2} \).

Step 2: Compare to the Harmonic Series

Notice that \( \frac{1}{n} \) is the basic harmonic series which is known to diverge. We are going to compare our series to this harmonic series. Let's express \( a_n = \frac{1}{n} (1 - \frac{1}{n}) \), so \( a_n \approx \frac{1}{n} \) for large \( n \) since \( \frac{1}{n^2} \) becomes negligible.

Step 3: Apply the Limit Comparison Test

Apply the Limit Comparison Test to \( \sum a_n \) and the harmonic series \( \sum \frac{1}{n} \). Compute the limit: \[ c = \lim_{n \to \infty} \frac{\frac{1}{n} - \frac{1}{n^2}}{\frac{1}{n}} = \lim_{n \to \infty} \left( 1 - \frac{1}{n} \right) = 1. \]Since \( c = 1 \) and finite (non-zero), the series \( \sum a_n \) behaves the same as the harmonic series, which diverges.

Step 4: Conclusion about Convergence

Since the harmonic series \( \sum \frac{1}{n} \) diverges and our limit comparison constant \( c = 1 \) is finite and non-zero, we conclude that the given series \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) \) also diverges.

Key Concepts

Harmonic Series

The harmonic series is one of the most simple yet fascinating series in mathematics. It is expressed as:

• \( \sum_{n=1}^{\infty} \frac{1}{n} \)

This series is famous because, despite its simple form, it diverges. Divergence means that as you add more and more terms, the series grows without bounds.
The harmonic series starts with 1 + 1/2 + 1/3 + 1/4 + ... and so on. Even though the terms get smaller, they don't decrease fast enough for the series to sum to a finite value. The concept of divergence in the harmonic series surprises many because the series terms become tiny, yet the sum heads to infinity.

Limit Comparison Test

The Limit Comparison Test is a powerful method in determining whether an infinite series converges or diverges. It involves comparing a given series to a known benchmark series.
Here's how it works:

• Given two series \( \sum a_n \) and \( \sum b_n \), compute the limit \( c = \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If \( c \) is a positive finite number, both series either converge or diverge together.

In this article's exercise, we used the limit comparison test to compare our series with the harmonic series. The result, \( c = 1 \), indicated that our series had the same behavior as the divergent harmonic series.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. You can think of it as endlessly adding numbers. Mathematically, it is expressed as \( \sum_{n=1}^{\infty} a_n \).
In the context of calculus and mathematical analysis, one of the main questions is whether this endless sum "converges" to a tangible number or "diverges" and grows indefinitely. When discussing infinite series, the terms initially considered often dictate the type of mathematical tests and methods applied to determine convergence or divergence.

Divergence

Divergence occurs when the sum of an infinite series does not approach a finite number. If a series diverges, it means that as we add more terms, the sum continues to grow towards infinity or does not settle at any single value.
To determine divergence, mathematicians use tests like the Harmonic Series Test or Limit Comparison Test. In our exercise, since the series we analyzed using the Limit Comparison Test shared characteristics with the harmonic series, it was shown to diverge as well. Divergence is a crucial concept in understanding the behavior of different series.

Mathematical Analysis

Mathematical analysis is like the detective work of mathematics. It involves uncovering the deeper properties and behaviors of mathematical objects such as functions and series. This branch deals with limits, sequences, series, and the nature of numbers and functions.
In our context, mathematical analysis helps us understand how to rigorously determine the convergence or divergence of infinite series. It equips us with tools like the Limit Comparison Test and guides our understanding of series like the harmonic series. Overall, mathematical analysis brings precision and clarity to problems about series convergence and divergence.