Divergence Test Study Notes

Divergence Test Overview

• The test for Divergence, also known as the Divergence Test, is a method used to determine the divergence of a series quickly.

• It is particularly useful as it saves time by immediately identifying that a series is divergent without requiring rigorous convergence proofs.

Statement of the Divergence Test

• The Divergence Test states:

  • If the limit of the sequence of terms $a<em>n$ does not exist, or if it exists but is not equal to zero, then the series $\sum a</em>n$ is a divergent series.

  • Consequently, the only scenario under which a series could converge is if its terms converge to zero.

Proof of the Divergence Test

• To prove this test, we will demonstrate an equivalent statement:

  • If a series is convergent, then the limit of its terms must be zero.

Understanding Equivalence

• It’s critical to recognize why these statements are equivalent:

  • Divergence Test: If the limit of $a_n$ is not zero (either it doesn’t exist or equals a non-zero number), then the series diverges.

  • Convergence Implication: If the series converges, it must mean that the terms approach zero; otherwise, the series cannot converge.

Setup for Proof

• Assume we have a convergent series represented by the summation of terms $a_n$.

• Let $S_n$ represent the nth partial sum of the series, defined as:

  • $S<em>n = a</em>1 + a<em>2 + … + a</em>n$

• The nth term can be expressed as:

  • $a<em>n = S</em>n - S_{n-1}$

Convergence of Partial Sums

• Since the series converges, the sequence of partial sums $S_n$ is also convergent.

• By definition:

  • $\lim<em>{n \to \infty} S</em>n = S$

• Also:

  • $\lim<em>{n \to \infty} S</em>{n-1} = S$

• This can be established since $S<em>{n-1}$ is just one term prior to $S</em>n$, and as n approaches infinity, both limits approach the same value S.

Finding the Limit of Sequence Terms

• Given the equation for $a_n$, we can find:

  • $\lim<em>{n \to \infty} a</em>n = \lim<em>{n \to \infty} (S</em>n - S_{n-1})$

• By applying the limit properties, this becomes:

  • $\lim<em>{n \to \infty} a</em>n = \lim<em>{n \to \infty} S</em>n - \lim<em>{n \to \infty} S</em>{n-1}$

• Substituting the limits we established earlier yields:

  • $\lim<em>{n \to \infty} a</em>n = S - S = 0$

Conclusion of Proof

• We have shown that if a series converges, then the limit of its terms must equal zero:

  • $\lim<em>{n \to \infty} a</em>n = 0$

• Therefore, according to the Divergence Test:

  • If the limit of the terms does not equal zero, the series must be divergent.

Important Note on Converse

• It is vital to note that the converse of this statement is not true:

  • If we know only that the limit of the terms approaches zero, this does not imply that the series converges.

  • Example: The limit of $\frac{1}{n}$ as $n$ approaches infinity is zero, however, the sum of all these terms (the harmonic series) diverges.

Summary of Divergence Test Applications

• If the limit of a sequence approaches zero, we cannot definitively conclude whether the series converges or diverges; the series may do either.

• However, if the limit of the terms does not approach zero, then we can assert that the series is divergent.