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Engineering Mathematics
book

Engineering Mathematics

by Babu Ram
June 2009
Intermediate to advanced content levelIntermediate to advanced
1100 pages
42h 22m
English
Pearson India
Content preview from Engineering Mathematics
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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Publisher Resources

ISBN: 9789332515611