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:

    “Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
    —Jean Baudrillard (b. 1929)

    “If a test of civilization be sought, none can be so sure as the condition of that half of society over which the other half has power.”
    —Harriet Martineau (1802–1876)