Question

If \(\lim n \rightarrow \infty a_{n}=c \neq 0\) or \(\lim n \rightarrow \infty a_{n}\) does not exist, then the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.

Short answer

The series diverges if \\(\lim_{n\to\infty} a_n\neq 0\\) or does not exist.

Step-by-step solution

Understanding Infinite Series

The series \(\sum_{n=1}^{\infty} a_n\) is an infinite sum of terms \(a_n\). To determine if it converges or diverges, we examine the behavior of its terms as \(n\) approaches infinity.

Convergence of a Series

For a series \(\sum_{n=1}^{\infty} a_n\) to converge, \(\lim_{n \rightarrow \infty} a_n\) must equal 0. This means that the terms of the series must become arbitrarily small as \(n\) increases.

Applying the Test for Divergence

If \(\lim_{n \rightarrow \infty} a_n = c eq 0\), then the terms of the series do not approach zero, implying the series cannot converge. This is because the sum of significant terms does not stabilize to a finite number.

Case of Non-Existence

If \(\lim_{n \rightarrow \infty} a_n\) does not exist, it means the terms of the series fluctuate indefinitely and do not settle down to a specific value, which also prevents convergence.

Conclusion on Divergence

For both cases, where \(\lim_{n \rightarrow \infty} a_n = c eq 0\) or \(\lim_{n \rightarrow \infty} a_n\) does not exist, the series \(\sum_{n=1}^{\infty} a_n\) must diverge.

Key concepts

Convergence

When we talk about the convergence of an infinite series like \( \sum_{n=1}^{\infty} a_n \), we're essentially looking to see if as you add up all the terms, the sum approaches a certain finite number. This is a fundamental concept in mathematical analysis and is crucial for understanding whether an infinite series can "settle down" to a fixed sum.
For a series to converge, it requires that the sequence of its partial sums, \( S_n = a_1 + a_2 + \cdots + a_n \), approaches a specific limit as \( n \) goes to infinity.

• One of the simplest tests to check for convergence is ensuring that the terms \( a_n \) of the series approach zero.
• If \( \lim_{n \to \infty} a_n = 0\), then the series could potentially be convergent, but further tests are usually needed to confirm this.

Understanding convergence helps us ascertain if the series resolves into a specific number, indicating a stable and predictable result.

Divergence

Divergence occurs when an infinite series does not settle down into a finite sum. This means as you add more and more terms, either the series grows indefinitely or fails to approach any particular value. Divergence can happen for various reasons, and identifying these is crucial in mathematical series analysis.

• If the terms \( a_n \) of the series do not approach zero as \( n \) increases, the series is guaranteed to diverge. This is because larger terms keep adding to the sum, preventing it from stabilizing.
• Moreover, if the limit \( \lim_{n \to \infty} a_n \) fails to exist, it indicates fluctuations in the series' behavior, meaning it doesn't settle down to a single value.

Divergence is an indication that the series is wild and has no finite resolution. This is the opposite of convergence, where everything comes together smoothly.

Test for Divergence

The test for divergence is a straightforward but powerful tool to determine if a series definitely does not converge. According to this test, for the series \( \sum_{n=1}^{\infty} a_n \) to converge, it is necessary that \( \lim_{n \to \infty} a_n = 0 \). If this condition is not satisfied, the series diverges.

• In cases where \( \lim_{n \to \infty} a_n eq 0 \), the terms are too large to allow the sum to stabilize, instantly indicating divergence.
• Alternatively, if \( \lim_{n \to \infty} a_n \) does not exist, it exhibits a lack of pattern or trend, again signaling divergence.

The test for divergence is often the first step employed to quickly rule out series that can't possibly converge, streamlining the analysis process for more complex series.