Study Notes on Infinite Series and Convergence Tests

Infinite Series and Absolute Convergence

Introduction to Series

  • Definition of a Series: A series is the sum of the terms of a sequence.
  • Example of Alternating Series:
    • Starts with terms such as:
    • First term: $- rac{1}{3}$ (negative)
    • Second term: $+ rac{1}{2}$ (positive)
    • Third term: $- rac{1}{4}$ (negative)
    • Fourth term: $+ rac{1}{5}$ (positive)
    • Pattern: Alternates between negative and positive.
    • General Form: If an element is denoted by $an$, then the series generally can be written as $an = (-1)^n rac{1}{n+1}$.

Absolute Convergence

  • Definition of Absolute Value: The absolute value of a term $an$, denoted $|an|$, is defined as:
    • For any negative term, the absolute value is its positive counterpart, e.g. $|- rac{1}{3}| = rac{1}{3}$.
  • Convergence of Absolute Value Series: When considering the absolute terms, the series to be considered is:
    • ext{Series: } ext{sum of } rac{1}{n^2}.
  • Conclusion of Absolute Convergence: The series made up of the absolute values ext{sum of } rac{1}{n^2} is recognized to be convergent.
  • Implication: If the series of absolute values converges, then the original series converges absolutely.

Absolute Convergence Test

  • The Test: If ext{sum of } |an| converges, then the series ext{sum of } an converges absolutely.
  • Importance: Proving absolute convergence aids in proving regular convergence directly through a theorem stating that absolute convergence leads to convergence of the original series.

Tests for Convergence

  • Types of Convergence Tests: The two primary tools for determining convergence of a series are:
    • Ratio Test
    • Root Test

Ratio Test

  • Definition: The ratio test states that for a series ext{sum of } a_n, the convergence can be tested by calculating:

    • ho = ext{lim as } n
      ightarrow ext{infinity} rac{|a{n+1}|}{|an|}.
  • Interpretation of $ ho$: Based on the value of ho:
    • If
      ho < 1, the series converges absolutely.
    • If
      ho > 1, or
      ho is infinite, the series diverges.
    • If
      ho = 1, the test is inconclusive.
  • Application:
    • Necessary to compute rac{a{n+1}}{an}.
    • If a_n = rac{n^{18}}{18^n}:
    • The next term would be a_{n+1} = rac{(n+1)^{18}}{18^{n+1}}.
    • Goal: Simplify and analyze the limit:
    • ext{lim as } n
      ightarrow ext{infinity} rac{(n+1)^{18}}{18^{n+1}} imes rac{18^n}{n^{18}}.

Simplifying the Limit

  • Guideline to Simplification: For any need to divide fraction, remember:
    • rac{a}{b} imes rac{c}{d} = rac{a imes c}{b imes d}.
  • Reduction Steps:
    • Simplify further by writing 18^{n+1} = 18^n imes 18 and reduce common bases, leading to:
    • rac{n}{n+1} (cancellation of like terms).
  • Final Limit Expression: Therefore, the limit reduces to:
    • ext{lim as } n
      ightarrow ext{infinity} rac{n^{18}}{(n+1)^{18}} = 1, leads to convergence.

Conclusion of Ratio Test

  • The outcome of the test indicates:
    • When
      ho < 1, it confirms that the series converges absolutely and also converges regularly.

Recursive Series and Root Test

  • Recursive Series: Describes a scenario where terms depend on previous terms, known as recursive series.
  • Root Test: Useful when handling terms where $n$ is exponentiated:
    • Determine:
    • ext{lim as } n
      ightarrow ext{infinity} ext{(n-th root of terms)} .
    • The root test works with expressions $n + a$, where $a$ is constant, in the numerator and subsequently evaluates limits based on factorial format or polynomial powers:
    • Transformation of general series behavior can be captured this way, proving convergences and divergences based on limit computation.