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Concept
Alternating series test
Summary
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.
The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.
A series of the form
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test guarantees that an alternating series converges if the following two conditions are met:
decreases monotonically, i.e., , and
Moreover, let L denote the sum of the series, then the partial sum
approximates L with error bounded by the next omitted term:
Suppose we are given a series of the form , where and for all natural numbers n. (The case follows by taking the negative.)
We will prove that both the partial sums with odd number of terms, and with even number of terms, converge to the same number L. Thus the usual partial sum also converges to L.
The odd partial sums decrease monotonically:
while the even partial sums increase monotonically:
both because an decreases monotonically with n.
Moreover, since an are positive, . Thus we can collect these facts to form the following suggestive inequality:
Now, note that a1 − a2 is a lower bound of the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even partial sum converges too.
Finally, they must converge to the same number because
Call the limit L, then the monotone convergence theorem also tells us extra information that
for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there is an odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.
Official source
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