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: $|\frac{a<em>{n+1}}{a</em>n}|$.
• 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: $x^{2n + 3} <br /> ightarrow x^{2n} \cdot x^{3}$.
• 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.