10/14 Alternating series

Overview of Series Convergence and Divergence

• This section will explore various types of series, particularly focusing on alternating series and the methods used to determine their convergence or divergence.

Types of Series Discussed

• Harmonic Series and Divergence

  • The harmonic series diverges.

• Series of the Form $\frac{n}{n^2}$ and Convergence

  • This can be rewritten as $\frac{1}{n}$, which diverges as it resembles the harmonic series.

• Series with Negative Terms

  • Issues arise when a series contains infinitely many negative terms.

  • Comparison tests become invalid if terms are not eventually non-negative.

Comparison Tests and Limitations

• Direct Comparison Test and Limit Comparison Test

  • These tests generally require series to be positive for applicability.

  • If there are infinite negative terms, comparisons become unreliable.

• Example Outcomes

  • Series resembling $\frac{1}{n}$ may diverge or converge based on the negativity of terms.

Alternating Series

• Definition of Alternating Series

  • Series of the form: $\sum (-1)^n a<em>n$ where $a</em>n$ is positive.

  • Alternates in sign (positive and negative terms).

• Significance of Alternation

  • The term $(-1)^n$ ensures positive and negative contributions alternately.

  • Essential that terms $a_n$ remain positive to apply specific tests.

Alternating Series Test (AST)

• Conditions for Convergence

  • If $\lim<em>{n \to \infty} a</em>n = 0$ and a{n+1} < an for all $n$ beyond some point, the series converges.

  • This test applies specifically to alternating series, providing a straightforward check for convergence without requiring comparison with other series.

• Example Application

  • For the series $\sum (-1)^{n+1} \frac{1}{n}$ (alternating harmonic series):

    • Check that $\lim \frac{1}{n} = 0$ is true and that the sequence is decreasing.

    • Thus, this series converges by the AST.

Error Estimation in Convergence

• Remainder Theorem for Alternating Series

  • This theorem allows for bounding the error when approximating the sum of an alternating series.

  • The error made from truncating the series after $n$ terms is at most the absolute value of the next term, $|a_{n+1}|$.

  • Example: If you sum to a certain term, the error is confined to the magnitude of the next term.

Convergence Types: Absolute vs. Conditional

• Absolute Convergence

  • A series converges absolutely if the series of its absolute values converges.

  • If a series converges absolutely, it will also converge, and the absolute convergence implies that any rearrangement of terms will also converge.

• Conditional Convergence

  • A series that converges but does not converge absolutely is conditionally convergent.

  • Example: The alternating harmonic series is conditionally convergent as it converges but the harmonic series diverges when all terms are treated as positive.

• Important Definitions

  • Converges Absolutely: If $\sum |a_n|$ converges.

  • Converges Conditionally: If $\sum a<em>n$ converges but $\sum |a</em>n|$ diverges.

Rearrangement Theorem

• A significant result indicating that for conditionally convergent series, rearranging terms can yield different sums or even cause the series to diverge.

• Significance: This highlights caution when manipulating series that do not absolutely converge.

Conclusion and Practical Considerations

• The alternating series test provides a simpler verification method for convergence, especially in settings with alternating terms.

• Understanding absolute vs. conditional convergence is crucial for deeper insights into series behavior, implications for rearrangement, and overall mathematical analysis for problem-solving.

• Continuous practice with numerous examples will help solidify concepts, particularly regarding convergence tests and estimating error in practical applications.