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:
  • $ext{Series: } ext{sum of } \frac{1}{n^2}$.
• Conclusion of Absolute Convergence: The series made up of the absolute values $ext{sum of } \frac{1}{n^2}$ 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 $ext{sum of } |a<em>n|$ converges, then the series $ext{sum of } a</em>n$ 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 $\frac{a<em>{n+1}}{a</em>n}$.
  • If $a_n = \frac{n^{18}}{18^n}$:
  • The next term would be $a_{n+1} = \frac{(n+1)^{18}}{18^{n+1}}$.
  • 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:
  • $\frac{a}{b} imes \frac{c}{d} = \frac{a imes c}{b imes d}$.
• Reduction Steps:
  • Simplify further by writing $18^{n+1} = 18^n imes 18$ and reduce common bases, leading to:
  • $\frac{n}{n+1}$ (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.