Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 26 :  Conditional convergence [Section 26.2]
26.2.6 Theorem (Alternating series test):
 

Let be an alternating series such that
(i)
(ii)Then is convergent.

  Proof
 

Let
            .
To show that is convergent, we will show that
            .
Since is alternating, let us suppose
            
(The proof for the other case is similar). Firstly we observe that since,
            
is an increasing sequence bounded above by and
            
implies that is decreasing sequence, bounded below by 0. Hence, both and are convergent sequences. Let
            
Since
            
and we have . This completes the proof.

 
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