proof of absolute convergence theorem

Suppose that $\sum a_{n}$ is absolutely convergent, i.e., that $\sum|a_{n}|$ is convergent. First of all, notice that

0\leq a_{n}+|a_{n}|\leq 2|a_{n}|,

and since the series $\sum(a_{n}+|a_{n}|)$ has non-negative terms it can be compared with $\sum 2|a_{n}|=2\sum|a_{n}|$ and hence converges.

On the other hand

\sum_{n=1}^{N}a_{n}=\sum_{n=1}^{N}(a_{n}+|a_{n}|)-\sum_{n=1}^{N}|a_{n}|.

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 $\sum a_{n}$ 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