Question

\(\mathrm{T} / \mathrm{F}:\) If \(\left\\{a_{n}\right\\}\) converges to \(0,\) then \(\sum_{n=0}^{\infty} a_{n}\) converges.

Short answer

False, a sequence converging to 0 does not ensure series convergence.

Step-by-step solution

- Understand the Given Situation

The problem states that a sequence \( \{a_n\} \) converges to 0. It means that as \( n \) becomes very large, the elements \( a_n \) approach 0.

- Review Convergence of Series

A series \( \sum_{n=0}^{\infty} a_n \) is said to converge if the sequence of its partial sums \( S_N = a_0 + a_1 + a_2 + \cdots + a_N \) has a finite limit as \( N \to \infty \).

- Apply the Necessary Condition for Series Convergence

A necessary condition for the convergence of the series \( \sum_{n=0}^{\infty} a_n \) is that the sequence \( a_n \to 0 \) as \( n \to \infty \). However, this condition alone is not sufficient to guarantee series convergence.

- Consider Counterexamples

One common counterexample is the harmonic series, where \( a_n = \frac{1}{n} \). Here, \( a_n \to 0 \), but \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges. This shows that having \( a_n \to 0 \) does not ensure the convergence of \( \sum_{n=0}^{\infty} a_n \).

- Draw a Conclusion

Based on the counterexample, we see that just because a sequence converges to 0, it does not mean that the series formed by summing its terms will converge. Therefore, the given statement is not true.

Key concepts

Understanding Sequence Convergence

In mathematics, a sequence is an ordered list of numbers. A sequence converges if its terms approach a specific value as the sequence extends indefinitely. For instance, the sequence \( \{a_n\} \) converges to 0 when, as \( n \) becomes very large, the terms \( a_n \) get arbitrarily close to 0. This behavior is fundamental in series analysis, as it often indicates how the terms behave over time. To check sequence convergence, you can consider whether there exists a number (limit) that the terms tend towards and remain close to for all sufficiently large indices \( n \). If this is the case, the sequence is said to converge.

Exploring Partial Sums

When discussing series, the concept of partial sums comes into play. A series \( \sum_{n=0}^{\infty} a_n \) consists of the sum of its terms from a sequence. However, to examine if a series converges, one must look at the sequence of its partial sums. If we denote the \( N^{th} \) partial sum as \( S_N = a_0 + a_1 + a_2 + \cdots + a_N \), then the series converges if the sequence of these partial sums approaches a finite limit as \( N \to \infty \).
This is akin to watching how the accumulated value behaves over the infinite horizon. If this accumulated amount approaches a specific number, the series is said to converge. Otherwise, if the sum keeps growing without bound, the series diverges.

Examining the Harmonic Series

The harmonic series is a classic example in the study of convergence. It is the series \( \sum_{n=1}^{\infty} \frac{1}{n} \). Each term \( a_n = \frac{1}{n} \) indeed converges to 0 as \( n \to \infty \), which might suggest convergence. However, the harmonic series itself is known to diverge.
This means the partial sums \( S_N = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{N} \) do not approach a finite limit but instead grow without bound. The behavior of the harmonic series verifies the importance of not confusing sequence and series convergence. Just because the terms in a sequence approach zero doesn't imply the sum of those terms will stabilize.

Necessary Conditions for Convergence

When analyzing series convergence, a crucial starting point is the sequence \( a_n \) converging to 0. This is a necessary condition. Without this limit going to zero, a series can't converge. However, it is not sufficient on its own. Many series with terms converging to 0 actually diverge, like the harmonic series.
Additional conditions or convergence tests, such as the Comparison Test, Ratio Test, or Integrals Test, are often required to conclusively determine a series' convergence. Understanding this helps in correctly interpreting whether a series converges, and avoids assumptions based solely on initial term behavior. Thus, approaching series analysis through multiple conditions and tests is vital for accurate conclusions.