Question

True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \lim _{n \rightarrow \infty} a_{n}=0, \text { then } \sum_{n=1}^{\infty} a_{n} \text { converges } $$

Short answer

False. The harmonic series is a counterexample.

Step-by-step solution

Understanding the Statement

We are given the statement: "If \( \lim_{n \to \infty} a_n = 0 \), then \( \sum_{n=1}^{\infty} a_n \) converges." We need to determine if this statement is true or false by providing a proof or a counterexample.

Review Series and Sequence Convergence

To prove or disprove the statement, remember two key points: 1) A series \( \sum_{n=1}^{\infty} a_n \) converges if the partial sums approach a finite limit. 2) One necessary (but not sufficient) condition for the convergence of a series is that the terms \( a_n \) must approach zero, i.e., \( \lim_{n \to \infty} a_n = 0 \). However, this does not guarantee convergence.

Counterexample - Harmonic Series

Consider the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). Here, \( a_n = \frac{1}{n} \) and we have \( \lim_{n \to \infty} \frac{1}{n} = 0 \). However, it is a known fact that the harmonic series diverges.

Conclusion

Since the harmonic series serves as a counterexample, it shows that even if \( \lim_{n \to \infty} a_n = 0 \), the series \( \sum_{n=1}^{\infty} a_n \) does not necessarily converge. Thus, the given statement is false.

Key concepts

Series Convergence

In mathematics, the concept of series convergence is pivotal in understanding how infinite sums behave. A series is the sum of the terms of a sequence. For a series \( \sum_{n=1}^{\infty} a_n \) to converge, the sequence of its partial sums must tend towards a finite limit as you add more and more terms.
To visualize this, consider adding fractions of smaller and smaller values. If, as you add more terms, the total stabilizes at a specific number, we say the series converges.

• If the partial sums grow without bound or oscillate indefinitely, the series diverges.
• Convergence requires that the partial sums \( S_n = a_1 + a_2 + \cdots + a_n \) must approach a fixed value as \( n \to \infty \).
• It is not enough that the terms \( a_n \) themselves shrink to zero.

Even if each sequence term becomes negligible, the cumulative effect might not limit-bound, thereby diverging. One illustrative way to think about it is by comparing the series with a bowl. If each term is a small amount of liquid, convergence means the bowl eventually doesn’t overflow.

Harmonic Series

The harmonic series is an essential example when discussing series convergence due to its divergence, despite having terms approaching zero. It is defined as \( \sum_{n=1}^{\infty} \frac{1}{n} \). While each term individually becomes smaller and smaller, approaching zero, the accumulated total never settles at a finite value; instead, it grows indefinitely.
This intriguing behavior demonstrates why merely having terms that tend to zero (as \( n \to \infty \)) isn't enough to ensure the convergence of a series.

• The divergence of the harmonic series can be quite surprising and is a classic example used to illustrate this crucial point about convergence.
• Mathematically, the harmonic series satisfies the condition \( \lim_{n \to \infty} \frac{1}{n} = 0 \), but it sums to an infinite amount.

The harmonic series teaches us that we must consider more than just the behavior of individual terms. It shows that convergence analysis requires looking at the sum as a whole, not just at the limit of its sequence terms.

Sequence Limits

The concept of sequence limits is fundamental in calculus and mathematical analysis, shaping our understanding of series behavior and convergence.
A sequence limit entails the value that the terms of a sequence \( a_n \) approach as \( n \to \infty \). When dealing with series, sequence limits help determine a necessary condition for convergence—whether the terms of the series approach zero.

• Mathematically, if \( \lim_{n \to \infty} a_n = L \), it means that as \( n \) becomes very large, \( a_n \) gets arbitrarily close to \( L \).
• It’s crucial to understand that having \( a_n \to 0 \) is a necessary condition for convergence but not a sufficient one.

This distinction is what we see with the harmonic series, where \( \lim_{n \to \infty} \frac{1}{n} = 0 \) but the series still diverges. Therefore, sequence limits provide essential insights, yet they are just one piece of the convergence puzzle. Understanding the total behavior of a series requires examining both the sequence limits and the nature of partial sums.