BC Calculus - Unit 10 Packet: Infinite Series (Part 1)
Overview of Infinite Series
Tests for Convergence: Several tests are used to determine the convergence of series: nth term test, geometric series test, p-series, integral test, direct and limit comparison tests, ratio test, and root test.
10.1 Convergent & Divergent Infinite Series
Definition of Sequence: A sequence is a collection of numbers in one-to-one correspondence with positive integers. For example, ( an = egin{cases} 1, 5, 7, 17, 31, ext{…} \ -2, 4, 2/0, 26, 80, 242, 6, 24, 120 \ ext{etc.} \ ext{This is not a regular sequence representation.} \ ext{Correct format: } an = 1 + (-2)^n ext{ showing terms for } n = 0, 1, 2, … \ \ ext{Result: Sequence: 1, -1, 5, 7, …, etc.}
\end{cases} )Monotonic Sequences: A sequence is monotonic if it either
Never decreases: for all ( n, a{n+1} \ge an )
Never increases: for all ( n, a{n+1} \le an )
Bounded Sequences: A sequence is bounded if there's some upper and lower limit. Notation:
( a_n \le M ) (upper bound)
( a_n \ge N ) (lower bound)
A sequence is considered bounded if both conditions are satisfied.
Infinite Series: Represents the sum of the terms of a sequence.
Partial Sum: ( Sn = a1 + a2 + … + an )
Denotes the series as follows: ( S = extstyle igcup{n=1}^∞ an )
10.1.1 Convergence and Divergence of Series
For an infinite series ( extstyleigcup{n=1}^∞ an ), the sequence of partial sums ( Sn = a1 + a2 + … + an ). If ( ext{lim}{n o ext{∞}} Sn = S ), then the series converges, and ( S ) is the sum of the series.
Conversely, if the limit does not exist (i.e., diverges), then the series does not converge.
Example 1: Does the series converge or diverge?
Series: ( extstyleigcup_{n=1}^{∞} rac{2n}{n^2 + 2n} ).
Example 2: Calculating partial sums: Use a calculator to find ( S_n ) for specified values (e.g., 200, 1000).
Practices
1. Find Sequence of Partial Sums
Given the series ( extstyleigcup{n=1}^{∞} (-1)^n ), find ( S1, S2, S3, S4, S5 ).
2. Evaluate Partial Sums for a Series
Find ( Sn ) for ( extstyleigcup{n=1}^{∞} rac{1}{n^2} ).
3. Determine Convergence
If ( S_n = (-1)^{n+1} ) for n ≥ 1, what is the sum of the series?
Series representation must be transformed into general form to identify converging properties.
10.2 Notes - Geometric Series
Definition: A geometric sequence has a constant ratio between successive terms.
nth Term of a Geometric Sequence:
General form: ( an = a1 r^{n-1} )
Example: For the sequence ( 3, 6, 12, 24, 48,… ), where common ratio ( r = 2 ).
Convergence Criteria for Geometric Series:
Converges if ( |r| < 1 ), diverges if ( |r|
ightarrow 1 )
Sum Formula for Infinite Geometric Series:
If ( |r| < 1 ), the sum S is given by ( S = rac{a_1}{1 - r} ).
Practice Problems for Geometric Series
Find the value of the infinite geometric series ( S = 25 + 5 + 1 + rac{1}{5} +… ).
Calculate the geometric series sum when the first term and common ratio are given.
10.3 Notes - nth term, p-series, and Integral Test
Nth Term Test for Divergence
If ( ext{lim}{n o ext{∞}} an \neq 0 ), then the series ( extstyleigcup{n=1}^{∞} an ) diverges.
If ( ext{lim}{n o ext{∞}} an = 0 ), the test is inconclusive, further analysis is required.
Integral Test for Convergence
If ( f(x) ) is a positive, continuous, and decreasing function for ( x ext{ ≥ k} ), then:
( extstyleigcup{n=1}^{∞} an ) converges if ( extstyle igint_{k}^{∞} f(x) dx ) converges.
Practice Problems: Analyze series using the integral test and provide explanations for your conclusions.
p-Series Convergence Criteria
( extstyle igcup_{n=1}^{∞} rac{1}{n^p} )
Converges if ( p > 1 )
Diverges if ( p ≤ 1 )
10.4 Comparison Tests for Convergence
Direct Comparison Test
Let ( 0 < an < bn ). If ( Σ bn ) converges, then ( Σ an ) converges.
Limit Comparison Test
If ( extstyleigcup{n=1}^{∞} an ) and ( extstyleigcup{n=1}^{∞} bn ) converge or diverge, they're related.
Example Problem: Define direct or limit comparison tests and confirm convergence/divergence for specified series.
Additional Convergence Testing Methods
Behavior; determine if the series behaves like one of the known tests (geometric, p-series).
Use ratio and root tests where appropriate for factorial and exponentials series.
Summary of Tests for Infinite Series: Convergence Tests
These principles are foundational in evaluating infinite series for convergence or divergence. Apply tests systematically, paying attention to the structure of the series to deduce results regarding convergence and to prepare for higher-level topics in calculus.
Test Prep for Each Section
Include a variety of practice problems focusing on determining convergence/divergence using different tests.
Work through examples and ensure familiarity with key definitions, properties, and example problems involving Taylor series, power series, and sequence limits.