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Unformatted text preview: Series CheatSheet 1 Definitions and properties Definition 1.1. Let a n be a sequence of real numbers. Denote the partial sum of the first n elements in the sequence with S n : S n = n X i =1 a i The function that maps n to S n is a real sequence and we call it a series . If the sequence is convergent we denote the limit with: + X i =1 a i Proposition 1.2. Let a n be a sequence of non negative numbers. If the series S n = n i =1 a i is bounded from above its convergent. Proof. If all the summands a n are positive, the sequence S n is increasing. An increasing sequence which is bounded from above is convergent. Proposition 1.3. Let a n be a sequence and S n be the associated series. If S n is convergent, the sequence a n converges to zero. Remark 1.4 . The implication works in one direction only!!!! There are plenty of sequences a n that converge to zero, but their associated series are divergent. For example a n = 1 n . 2 Techniques to study the convergence 1st method: The following technique is useful to determine the convergence of some very basic series. We will call it the 2 k-method. Proposition 2.1. Let a n be a positive and decreasing sequence. The series + n =1 a n is convergent if and only if the series: + X k =0 2 k a 2 k is convergent. 2nd method: The following technique lets us compare a series with an improper integral. Some- times it might be easier to evaluate an improper integral than a series (most of the time this is not the case): 1 Proposition 2.2. Let f ( x ) be a real function on the interval [1 , + ) which is positive, continuous and decreasing. Let n be a natural number, the following inequality holds: Z n +1 1 f ( x ) dx n X i =1 f ( i ) Z n +1 1 f ( x ) dx + f (1)- f ( n ) The series + i =1 f ( i ) is convergent if and only if the improper integral R + 1 f ( x ) dx is con- vergent....
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