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# Absolute Convergence Test

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Prove the absolute convergence test:

Let the sum from n=m to infinity of a_n be a formal series of real numbers. If this series is absolutely convergent, then it is also conditionally convergent. Furthermore, in this case we have the triangle inequality - the absolute value of the sum from n=m to infinity of a_n <= the sum from n=m to infinity of the absolute value of a_n.

Let the sum from n=m to infinity of a_n be a formal series of real numbers. Then the sum from n=m to infinity of a_n converges if and only if, for every real number epsilon>0, there exists an integer N<=m such that the absolute value of the sum from n=p to q of a_n <= epsilon for all p, q>= N.

and

Triangle inequality for finite series: let m<=n be integers, and let a_i be a real number assigned to each integer m<= i <= n. Then we have
the absolute value of the sum from i=m to n of a_i <= the sum from i=m to n of the absolute value of a_i.