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 u_n is positive and monotonically decreases to the limit zero, then the alternating series u₁ – u₂+ u₃–u₄ + … is convergent.

Proof: Consider

 

u_n – u_n+1 + u_n+2 – ⋯ + (–1)^pu_n+p.

 

Writing this expression as

 

(u_n – u_n+1) + (u_n+2 – u_n+3) + ⋯,

 

we see that when p is odd, –u_n+p occurs in the last bracket; and when p is even, u_n+p is the last term. Since u_n ≥ u_{n+ 1}, each bracket is non-negative and so the the expression discussed earlier is nonnegative.

Now we write the same expression ...