Alternating Series - Alternating Series Test

Alternating Series Test

The theorem known as "Leibniz Test" or the alternating series test tells us that an alternating series will converge if the terms an converge to 0 monotonically.

Proof: Suppose the sequence converges to zero and is monotone decreasing. If is odd and, we obtain the estimate via the following calculation:

\begin{align} S_m - S_n & = \sum_{k=0}^m(-1)^k\,a_k\,-\,\sum_{k=0}^n\,(-1)^k\,a_k\ = \sum_{k=m+1}^n\,(-1)^k\,a_k \\ & =a_{m+1}-a_{m+2}+a_{m+3}-a_{m+4}+\cdots+a_n\\ & =\displaystyle a_{m+1}-(a_{m+2}-a_{m+3}) - (a_{m+4}-a_{m+5}) -\cdots-a_n \le a_{m+1}\le a_{m} \end{align}

Since is monotonically decreasing, the terms are negative. Thus, we have the final inequality .Similarly it can be shown that . Since converges to, our partial sums form a cauchy sequence (i.e. the series satisfies the cauchy convergence criterion for series) and therefore converge. The argument for even is similar.

Read more about this topic:  Alternating Series

Famous quotes containing the words series and/or test:

    “Autobiography is only to be trusted when it reveals something disgraceful. A man who gives a good account of himself is probably lying, since any life when viewed from the inside is simply a series of defeats.”
    —George Orwell (1903–1950)

    “He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
    —Herman Melville (1819–1891)