Question

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

Short answer

True, the series diverges if the limit is nonzero.

Step-by-step solution

Understand the Problem Statement

We are asked to determine if the statement, "If \( \lim_{n \rightarrow \infty} a_{n} eq 0 \), then \( \sum_{n=1}^{\infty} a_{n} \) diverges," is true or false. We need to justify our answer with a proof or a counterexample.

State the Limit Requirement for Series Convergence

A basic property of infinite series states that if \( \sum_{n=1}^{\infty} a_{n} \) converges, then \( \lim_{n \rightarrow \infty} a_{n} = 0 \). Conversely, if \( \lim_{n \rightarrow \infty} a_{n} eq 0 \), the series \( \sum_{n=1}^{\infty} a_{n} \) must diverge.

Provide Proof for the Statement

Assume \( \lim_{n \rightarrow \infty} a_{n} = c eq 0 \). Then there exists some \( \epsilon > 0 \) such that \( |a_n - c| < \epsilon \) for all sufficiently large \( n \). Given that \( a_n \) approaches the nonzero constant \( c \), the partial sums of \( \sum_{n=1}^{\infty} a_{n} \) do not settle to a finite value; the series cannot converge. Therefore, \( \sum_{n=1}^{\infty} a_{n} \)must diverge.

Explore Counterexample Possibility

Attempting to find a counterexample, consider an arbitrary series \( \sum_{n=1}^{\infty} a_{n} \) with \( \lim_{n \rightarrow \infty} a_{n} eq 0 \). Such as \( a_n = 1 \), which clearly doesn't lead to a convergent series due to the non-zero limit, confirming the original statement.

Key concepts

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Formally, if you have a sequence \( a_1, a_2, a_3, \ldots \), the series is written as \( \sum_{n=1}^{\infty} a_n \). This notation signifies that we are adding each term in the sequence forever. Understanding infinite series is crucial in mathematics as they can represent complex functions, help calculate limits, and solve equations in calculus.

However, not every infinite series will sum to a finite number. The series might not add up to anything meaningful if the terms do not get smaller fast enough. Determining whether an infinite series converges (adds up to a finite sum) or diverges (does not add up to a finite sum) is a key question in mathematical analysis. Often, techniques like the comparison test, ratio test, or integral test are used to analyze series behavior.

Convergence of Series

A series converges if the sequence of its partial sums \( s_N = a_1 + a_2 + \cdots + a_N \) approaches a finite limit as \( N \) becomes very large. This means that as you keep adding more terms from the series, the total sum gets closer and closer to a particular number.

• If the series converges, the value that the partial sums approach is called the sum of the series.
• Conversely, if the partial sums grow without bound or oscillate without settling to a point, the series diverges.

One quick test for determining the convergence of a series is the "nth-term test for divergence." If \( \lim_{n \to \infty} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) cannot converge. This is because, for a series to sum to a finite number, the terms need to become closer to zero—if they don't, the sum can't settle on a single finite value.

Limit of a Sequence

In the context of series, the limit of a sequence is about what the terms \( a_n \) approach as \( n \) goes to infinity. A sequence could converge to a limit if its terms become arbitrarily close to a single value. With only sequences where the terms approach zero, you may have a chance of forming a converging series from them.

When examining the series, the logical first step is to find \( \lim_{n \to \infty} a_n \). If the terms do not approach zero, it undoubtedly causes the series to diverge. In more mathematical terms, if \( \lim_{n \to \infty} a_n = L \, eq 0 \), the sum of these terms would continue to build without ever settling down to a single number.

• The key takeaway is: the limit of the sequence informs the behavior of the series.
• Thus, non-zero limits usually ensure the divergence of the corresponding series unless the terms specifically conform to any special formats or structures that allow convergence through other methods.