proof of absolute convergence theorem

Suppose that an is absolutely convergent, i.e., that |an| is convergent. First of all, notice that


and since the series (an+|an|) has non-negative terms it can be compared with 2|an|=2|an| and hence converges.

On the other hand


Since both the partial sums on the right hand side are convergent, the partial sum on the left hand side is also convergent. So, the series an is convergent.

Title proof of absolute convergence theorem
Canonical name ProofOfAbsoluteConvergenceTheorem
Date of creation 2013-03-22 13:41:52
Last modified on 2013-03-22 13:41:52
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 6
Author paolini (1187)
Entry type Proof
Classification msc 40A05