Study Notes for AP Calculus Unit 10 - Infinite Series

Infinite Series: Defined as the summation of a sequence with an infinite number of terms.
- Notation: If ( a_n ) is an infinite sequence, then an infinite series is represented as ( \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \ldots + a_n + \ldots )
Partial Sums: To determine the convergence of a series, we analyze the sequence of partial sums. If the sequence of partial sums converges, then the series converges as well.
- Example: For the series ( \sum_{n=1}^{\infty} \frac{1}{2^n} )
- Partial sums: 1/2, 3/4, 7/8, …\ ( S = \lim_{n \to \infty} S_n = \frac{1}{2} )
Convergence Conditions:
- If ( S_n ) converges to ( S ), then the series converges, and ( S ) is called the sum of the series.
- If ( S_n ) diverges, then the series diverges.

Geometric Series: A specific type of series defined by the form ( \sum_{n=0}^{\infty} ar^n )
- Convergence: A geometric series converges if ( |r| < 1 ) and diverges if ( |r| \geq 1 ).
- Sum: If the series converges, the sum is given by ( S = \frac{a}{1 - r} )
Example #2: Consider the series 8 + ( \frac{8}{5} + \frac{8}{25} + \frac{8}{125} + \ldots )
- Find:
- A: ( a_1 = 8 )
- B: Ratio ( r = \frac{1}{5} )
- C: Summation notation: ( \sum_{n=0}^{\infty} 8 \left(\frac{1}{5}\right)^n )
- D: ( S_4 = 8(1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125}) = \frac{40}{3} )
- E: Find ( S = \frac{8}{1 - \frac{1}{5}} = 10 )

Nth Term Test: If a series ( \sum_{n=1}^{\infty} a_n ) converges, then ( \lim_{n \to \infty} a_n = 0 ). If ( \lim_{n \to \infty} a_n \neq 0 ), then the series diverges.

Integral Test: Useful for determining the convergence of series with positive terms:
- Criteria: The function ( f(x) ) must be continuous, positive, and decreasing on the interval.
- If ( \int_{1}^{\infty} f(x) dx ) converges, then the series converges.
- If it diverges, then the series diverges.

Definition: A series is said to be a p-series if it is of the form ( \sum_{n=1}^{\infty} \frac{1}{n^p} )
- Convergence Conditions:
- Converges if ( p > 1 )
- Diverges if ( p \leq 1 )

Direct Comparison Test: Let ( 0 < a_n \leq b_n ).
- If ( \sum b_n ) converges, then ( \sum a_n ) also converges.
- If ( \sum a_n ) diverges, then ( \sum b_n ) also diverges.
Limit Comparison Test: For series with positive terms:
- If ( \lim_{n \to \infty} \frac{a_n}{b_n} = L > 0 ), then both series converge or diverge together.

Definition of Alternating Series: A series whose terms alternate in sign.
Convergence Conditions:
1. Every term ( a_n ) is positive.
2. The sequence of terms ( a_n ) is decreasing: ( a_{n+1} < a_n ).
3. ( \lim_{n\to\infty} a_n = 0 ).
Error Bound: The absolute value of the remainder ( |S - S_n| \leq a_{n+1} ) for convergent alternating series.

Ratio Test: Useful for series, especially those featuring factorials or exponentials:
1. If ( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1 ) then the series converges absolutely.
2. If ( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| > 1 ) or it is infinite, the series diverges.
3. If it equals 1, the test is inconclusive.
Root Test: Similar to the Ratio Test but focuses on the nth root.
Exact criteria and exploration of examples.

Definition of Taylor Polynomial: For ( f ) that has ( n ) derivatives at ( x = c ):
- ( P_n(x) = f(c) + f'(c)(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \ldots + \frac{f^{(n)}(c)}{n!}(x - c)^n )
- If ( c = 0 ), the series is a Maclaurin Series.
Use Cases: Approximate functions near a given point, such as ( f(x) = e^x, \, sin(x), \, cos(x) ).
Working with Series: Differentiate/integrate term-by-term, applying to known series.

Definition: A power series is of the form ( \sum_{n=0}^{\infty} a_n (x - c)^n ).
Finding Convergence: Using the Ratio Test to determine radius and interval of convergence.
Convergence Conditions: Series can converge at its center ( c ) and exhibit different behaviors beyond.

Apply the Integral Test, Comparison Tests, and alternating series convergence to various series.
Create Taylor and Maclaurin series for functions analyzing convergence, estimating sum differences, and finding errors.

These topics lay the ground for understanding convergence behaviors in infinite series and their practical applications in calculus.