1.19 ALTERNATING SERIES

A series in which positive and negative terms occur alternately is called an Alternating Series.

Regarding convergence behavior of an alternating series, we have the following theorem, known as Leibnitz's Rule.

Theorem 1.19.    (Leibnitz's Rule). If un is positive and monotonically decreases to the limit zero, then the alternating series u1u2+ u3u4 + … is convergent.

Proof: Consider

 

unun+1 + un+2 – ⋯ + (–1)pun+p.

 

Writing this expression as

 

(unun+1) + (un+2un+3) + ⋯,

 

we see that when p is odd, –un+p occurs in the last bracket; and when p is even, un+p is the last term. Since unun+ 1, each bracket is non-negative and so the the expression discussed earlier is nonnegative.

Now we write the same expression ...

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