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Calculus II Notes 12.2

# Calculus II Notes 12.2 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.2 12.2 Series If we try to add the elements of an infinite sequence { a k } we get the infinite series a k k = 1 = a k = a 1 + a 2 +…+ a k +… . Can we actually add infinitely many numbers? In Zeno’s most famous paradox, Achilles can never catch the tortoise, if the tortoise is given a head start, because the Greeks assumed that such a sum must be infinite. The argument is that, when Achilles reaches the place where the tortoise started, the tortoise has moved ahead to a new location. When Achilles reaches that location, the tortoise has moved ahead again, and so on, ad infinitum. Definition a k k = 1 lim n →∞ a k k = 1 n = lim n →∞ S n = S where S n = a k k = 1 n is the n th partial sum of the series. If the limit exists, we say the series converges . Otherwise, it diverges . S is called the sum of the series. Note: This is actually the limit of a sequence of partial sums S n { } . It is also common to begin the sequence at k = 0 , or even a negative number. Examples A 1 k ( k + 1) k = 1 = 1 k 1 k + 1 k = 1 is convergent since S n = 1 1 1 2 + 1 2

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