Convergence and Divergence of Series Definition of an Infinite Series: Given a sequence of numbers fa n g, an expression of the form a k = a + a 2 + a 3 + +a n + is an infinite series. The number a n is the nth term of the series. The sequence fs n g defined by S = a S 2 = a + a 2 S 3 = a + a 2 + a 3. S n = a + a 2 + a 3 + +a n = is the sequence of partial sums of the series, the number S n being the nth partial sum. If the sequence of partial sums converges to a it L, we say that the series converges and that its sum is L. In this case we also write a + a 2 + a 3 + +a n + = nx a k a k = L: If the sequence of partial sums of the series does not converge, we say that the series diverges. Tail End of a Series: If a k diverges, then so does 00 a k. Similarly, if k=50 b k converges, then so does 0 b k. Of course its sum will likely be different. Thus the tail end of a series is what really determines divergence and convergence you can ignore the beginning terms or even change them without affecting whether the series diverges or converges. We often write X a k if we do not care where the summation begins. Linearity of Convergent Series: If X a k and X b k converge and c is a constant, then X ca k = c X a k, and X (a k + b k ) = X a k + X b k. Multiple of a Divergent Series: If X a k diverges and c 6= 0, then X ca k diverges. Grouping: The terms of a convergent series can be grouped in any way (provided the order of the terms is maintained) and the new series will converge with the same sum as the original series.
Convergence and Divergence of some Basic Series nth Term Test for Divergence: (the series aren't necessarily basic, but the test is) Given a series X a n, we find that ffl if n! a n 6= 0, or if n! a n fails to exist, then the series diverges. ffl if n! a n = 0, then the series may converge or it may diverge the test is inconclusive. Geometric Series: (results follow from taking the it of the partial sums) A geometric series is a series of the form ar k = a + ar + ar 2 + ar 3 + in which a and r are fixed real numbers and a 6= 0. We refer to r as the ratio of the geometric series. ffl If jrj <, the series converges and has sum ffl If jrj, the series diverges. a r. p-series: (results follow from using the Integral Test and the nth Term Test) A p-series is a series of the form in which p is a fixed real number. ffl If p >, the series converges. ffl If p», the series diverges. X k p = + 2 p + 3 p + 4 p + Harmonic Series: (note that this is just a p-series with p = ) The series Collapsing Series: k = + 2 + 3 + + is called the harmonic series. This series diverges. 4 A series is said to be collapsing if all but a finite number of terms cancel out in the nth partial sum. Take the it of the partial sums to determine convergence or divergence. Finite Series: All finite series converge! To find the sum, you can always add up all the terms one-by-one, but this can take a long time. For geometric sums, it will speed things up if you know that a + ar + ar 2 + +ar n = a arn+. r 2
Convergence and Divergence of Series with Positive Terms Note that many of the following tests can be stated more generally. For instance, you can often change positive to nonnegative or decreasing to nonincreasing and still be able to apply the test. Also since only the tail end of a series affects its convergence or divergence, the conditions of each test do not have to be satisfied for all n. They merely have to hold for all n N where N can be any positive integer. When in doubt, look up the exact wording in your calculus book. Ratio Test: be a series with positive terms, and suppose that ffl if ρ <, the series converges. n! a n+ = ρ: a n ffl if ρ >, the series diverges. ffl if ρ =, the series may converge or it may diverge the test is inconclusive. nth Root Test: be a series with positive terms, and suppose that ffl if ρ <, the series converges. n! np a n = ρ: ffl if ρ >, the series diverges. ffl if ρ =, the series may converge or it may diverge the test is inconclusive. Limit Comparison Test: be a series with positive terms. ffl Test for convergence. If there is a convergent series X c n n! a X n <, then a n converges. c n with positive terms for which ffl Test for divergence. If there is a divergent series X d n with positive terms for which n! a X n > 0, then a n diverges. d n Simplified Limit Comparison Test: If the terms of the two series X a n and X b n are positive, and the it of a n =b n is finite and positive, then both series converge or both diverge. 3
Ordinary Comparison Test: Let X a k be a series with positive terms. ffl Test for convergence. If there is a convergent series X c k with a k» c k for all k, then the X series a k converges. ffl Test for divergence. If there is a divergent series X d k with positive terms such that a k d k X for all k, then the series a k diverges. Integral Test: Let f be a continuous, positive, decreasing function on the interval [; ) and suppose a k = f(k) for all positive integers k. the infinite series Z a k converges if and only if the improper integral f(x) dx converges. When the series converges, we can approximate its sum using that: Z f(x) dx» a k» a + Z f(x) dx Remark: This test can be generalized by replacing each occurrence of the integer with M, where M can be any positive integer. Bounded-Sum Test: A series X a k of positive terms converges if and only if its partial sums are bounded above. 4
Convergence and Divergence of Series with Positive or Negative Terms Alternating Series Test: An alternating series is a series of the form in which a k is positive for all k. ( ) k+ a k = a a 2 + a 3 a 4 + This series converges if both of the following additional conditions hold:. a k a k+ for all k. 2. k! a k = 0. To approximate the sum of the series, use the partial sum S n = less than or equal to a n+. nx ( ) k+ a k. Your error will be X X We say that a k converges absolutely if ja k j converges. A series which converges but does not converge absolutely is said to converge conditionally. Absolute Convergence Test: If X ja k j converges, then X a k also converges. Absolute Ratio Test: be a series with nonzero terms, and suppose that n! ja n+j = ρ: ja n j ffl if ρ <, the series converges absolutely (hence converges). ffl if ρ >, the series diverges. ffl if ρ =, the series may converge or it may diverge the test is inconclusive. Rearrangement Theorem: The terms of a series which converges absolutely can be rearranged without affecting either the convergence or the sum of the series. 5