Absolute and Conditional Convergence; The Ratio, Root, and Alternating Series Tests

Absolute and Conditional Convergence

Definition of Absolute Convergence

A series \( \sum a_n \) is said to converge absolutely (or is absolutely convergent) if the corresponding series of absolute values, \( \sum |a_n| \), converges.

• Absolute convergence is a stronger form of convergence. If a series converges absolutely, it also converges in the usual sense.

• Absolute convergence is important for determining the behavior of series with both positive and negative terms.

The Absolute Convergence Test

If \( \sum |a_n| \) converges, then \( \sum a_n \) also converges.

• This test allows us to conclude convergence of a series by examining the series of absolute values.

Conditional Convergence

A series that converges, but does not converge absolutely, is said to converge conditionally.

• Conditional convergence often occurs in alternating series, where the series converges, but the series of absolute values diverges.

The Ratio and Root Tests

The Ratio Test

The Ratio Test is a powerful tool for determining the convergence of series, especially those involving factorials, exponentials, or nth powers.

• Let \( \sum a_n \) be any series and suppose that \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \rho \).

• If \( \rho < 1 \), the series converges absolutely.

• If \( \rho > 1 \) or \( \rho = \infty \), the series diverges.

• If \( \rho = 1 \), the test is inconclusive.

The Root Test

The Root Test is especially useful for series where the nth term is raised to the nth power.

• Let \( \sum a_n \) be any series and suppose that \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = \rho \).

• If \( \rho < 1 \), the series converges absolutely.

• If \( \rho > 1 \) or \( \rho = \infty \), the series diverges.

• If \( \rho = 1 \), the test is inconclusive.

Alternating Series and Conditional Convergence

The Alternating Series Test (Leibniz Test)

The Alternating Series Test provides a criterion for the convergence of series whose terms alternate in sign.

• The series \( \sum_{n=1}^\infty (-1)^{n+1} u_n \) converges if:

  1. All \( u_n \) are positive.

  2. The sequence \( u_n \) is eventually nonincreasing: \( u_n \geq u_{n+1} \) for all \( n \geq N \) for some integer N.

  3. \( u_n \to 0 \) as \( n \to \infty \).

Figure 10.15: The partial sums of an alternating series that satisfies the hypotheses of Theorem 15 for N = 1 straddle the limit from the beginning.

Comparison: Harmonic vs. Alternating Harmonic Series

• Figure 10.16(a): The harmonic series diverges, with partial sums that eventually exceed any constant M.

• Figure 10.16(b): The alternating harmonic series converges to \( \ln 2 \approx 0.693 \).

The Alternating Series Estimation Theorem

This theorem provides an estimate for the error when approximating the sum of an alternating series by its partial sums.

• If the alternating series \( \sum_{n=1}^\infty (-1)^{n+1} u_n \) satisfies the three conditions of the Alternating Series Test, then for n \geq N, the partial sum \( s_n \) approximates the sum L of the series with an error whose absolute value is less than u_{n+1}, the absolute value of the first unused term.

• The sum L lies between any two successive partial sums s_n and s_{n+1}, and the remainder L - s_n has the same sign as the first unused term.

Rearrangement of Series

The Rearrangement Theorem for Absolutely Convergent Series

If a series is absolutely convergent, any rearrangement of its terms will converge to the same sum.

• This property does not hold for conditionally convergent series, where rearrangement can lead to different sums or even divergence.

Summary of Series Tests

• nth-Term Test for Divergence: If \( a_n \not\to 0 \), the series diverges.

• Geometric Series: \( \sum ar^n \) converges if |r| < 1.

• p-Series: \( \sum 1/n^p \) converges if p > 1.

• Series with Nonnegative Terms: Use the Integral Test, Comparison Tests, Ratio or Root Test.

• Series with Some Negative Terms: Test for absolute convergence using the Ratio or Root Test.

• Alternating Series: Use the Alternating Series Test.