Study Notes on Integral and Ratio Test in Series Convergence

Integral and Wages

  • The focus is on finding the integral and applying the ratio test.

Definition of the Ratio Test

  • The ratio test involves the absolute value of a sub n plus one divided by a sub n.
  • Definition: |\frac{a{n+1}}{an}| .
  • Example: For a sequence involving factorials, where:
    • a_n = n! x^n .

Simplifying the Terms

  • To find a sub n plus one, replace n with n plus one:
    • a_{n+1} = (n+1)! x^{n+1} .
    • Therefore, the terms are:
    • a_n = n! x^n
    • a_{n+1} = (n+1)! x^{n+1} .

Applying the Ratio Test

  • The test is based on the limit of the sequence.
  • Considering the expression:
    • | \frac{(n+1)! x^{n+1}}{n! x^n} | .
  • This simplifies to:
    • |(n+1) x| .
  • If the limit as n approaches infinity of this expression is greater than one, it indicates divergence.
    • Specifically: DIVERGENCE when the absolute value of x multiplied by infinity yields greater than one.

Convergence and Divergence

  • The series converges only at c:
    • where c represents the center of the power series.
  • The radius of convergence is noted as:
    • R = 0 when the series diverges.
  • Definition: A series converging at c implies:
    • Example: If x = 0 , the series converges at that point.

Implications of Divergence

  • If the series diverges everywhere except x = 0, x = 0 becomes the center point (c).
  • When looking at the convergence from -\infty to \infty , the radius remains zero due to divergence.

Distance and Interval of Convergence

  • The radius (R) is the half distance from the center point.
    • If intervals are defined as parentheses (indicating divergence) or brackets (indicating convergence).
    • Example: Given intervals [1,3] or [-1,2], these would represent divergence, therefore a parenthesis would be used.

Application of the Ratio Test to New Series

  • Example series defined as:
    • a_n = 3(x - 2)^n .
  • The analysis follows with:
    • Cancelling the 3 factors in the ratio test expression, leaving:
    • |(x - 2)| .
  • Ending conditions:
    • Divergence occurs when endpoints are not included. Thus using parentheses instead of brackets.

Further Checks on the Interval

  • Always check for the interval of convergence which requires examining conditions:
    • The intervals must satisfy the criteria of being less than one regardless of being negative or positive.

Understanding Center and Series Characteristics

  • The center is further analyzed through raised functions:
    • Example: x^{2n + 3}
      ightarrow x^{2n} \cdot x^{3} .
  • Indices should be correctly labeled, e.g., n = 0 to infinity.

Conclusion on Calculations and Results

  • Further examples can examine the total length of intervals, calculate distances per specific criteria by subtracting:
    • Example: If given intervals [2,3],:
    • 3 - 2 = 1 then halving this gives you the final interval guidelines.
  • Key operations should always imply movement towards understanding the distance and the radius of convergence relative to the center of the series.