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Transcript
Calc 2 Lecture Notes Section 8.2 Page 1 of 2 Section 8.2: Infinite Series Big idea: Sometimes when you add up an infinite set of numbers, you get a finite answer. For this to happen, most of the numbers in that set had better be pretty darn close to zero. Thus, if the terms in a sum do not tend to zero, the sum will diverge. Big skill:. You should be able to compute the sum of an infinite geometric series, and use some basic tests to determine when a series diverges. Definition of an Infinite Series: n ak lim ak lim Sn S n k 1 k 1 n Practice: 0.1 k k 1 k k 1 1 k k 1 k 1 Theorem 2.1: Sum of an Infinite Geometric Series. a For a 0, the geometric series ar k converges to if r 1 and diverges if r 1 . The 1 r k 0 number r is sometimes called the common ratio or just the ratio. Practice: k 1 3 k 0 2 2 1.1 k k 0 0.1 k 1 k Calc 2 Lecture Notes Section 8.2 Page 2 of 2 Theorem 2.2: If you add up an infinite number of numbers and the sum doesn’t blow up, then the numbers must be really small numbers. If a k 1 k converges, then lim ak 0 . k kth-Term Test for Divergence. (Contrapositive of Theorem 2.2) If lim ak 0 , then k a k 1 k diverges. Practice: k 1 k k 0 Theorem 2.3: Combinations of Series (are what you think they’d be). If ak converges to A, and k 1 bk converges to B, then the series k 1 a k 1 k bk A B , and ca cA for any constant c. k k 1 If ak diverges, and k 1 bk diverges also, then the series k 1 a k 1 k bk diverges as well.