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Intro Real Analysis, Lec 24: Convergence of Series, Geometric Series, p-Series Test, Divergence Test
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Introduction to Real Analysis, Lecture 24.
(0:00) Career night announcement. Exam 3 date and topic coverage.
(1:21) Definition of an infinite series (infinite sum) and what it means for an infinite series to converge (the series converges if and only if the corresponding sequence of partial sums converges).
(6:04) A convergent geometric series example. Be comfortable with both summation (sigma) notation and writing it out. Be able to identify the common ratio r. It sums to a/(1 - r), which for this example is 21/5 = 4.2.
(11:14) General derivation of the formula for the sum of a convergent geometric series (start by deriving the formula for the nth partial sum). The series converges if and only if r is strictly between -1 and 1.
(20:19) Definition of a p-series and a description of when they converge (when p is greater than 1) and diverge (when p is less than or equal to 1). When p = 1 it is the harmonic series. The harmonic series diverges by the integral test or by considering appropriate inequalities for special subsequences of the sequence of partial sums.
(30:11) Divergence test (If a series converges, then its terms must converge to zero. The contrapositive is: if the terms of a series do not converge to zero, then the series diverges). Make sure you distinguish between the sequence of terms and the sequence of partial sums.
(35:39) "Proof" that 0 = 1 (zero equals one proof).
(38:34) Cauchy criterion for convergence of an infinite series.
(42:32) For a series of non-negative terms, the series will converge if and only if its sequence of partial sums is bounded above (since the sequence of partial sums will be monotone increasing).
(46:13) Algebraic combinations (sums, differences, constant multiples) of convergent series are convergent (and converge to the corresponding algebraic combination of the sums of the original series). The proofs are related to corresponding properties of algebraic combinations of sequences.
Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN.
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