Editing Alternating series (section)

== Approximating sums ==
The estimate above does not depend on <math>n</math>. So, if <math>a_n</math> is approaching 0 monotonically, the estimate provides an [[error bound]] for approximating infinite sums by partial sums:
<math display="block">\left|\sum_{k=0}^\infty(-1)^k\,a_k\,-\,\sum_{k=0}^m\,(-1)^k\,a_k\right|\le |a_{m+1}|.</math>That does not mean that this estimate always finds the very first element after which error is less than the modulus of the next term in the series. Indeed if you take <math>1-1/2+1/3-1/4+... = \ln 2</math> and try to find the term after which error is at most 0.00005, the inequality above shows that the partial sum up through <math>a_{20000}</math> is enough, but in fact this is twice as many terms as needed. Indeed, the error after summing first 9999 elements is 0.0000500025, and so taking the partial sum up through <math>a_{10000}</math> is sufficient. This series happens to have the property that constructing a new series with <math>a_n -a_{n+1}</math> also gives an alternating series where the Leibniz test applies and thus makes this simple error bound not optimal. This was improved by the Calabrese bound,<ref>{{Cite journal |last=Calabrese |first=Philip |date=March 1962 |title=A Note on Alternating Series |url=https://www.jstor.org/stable/2311056 |journal=The American Mathematical Monthly |volume=69 |issue=3 |pages=215–217 |doi=10.2307/2311056|jstor=2311056 }}</ref> discovered in 1962, that says that this property allows for a result 2 times less than with the Leibniz error bound. In fact this is also not optimal for series where this property applies 2 or more times, which is described by [[Richard Johnsonbaugh|Johnsonbaugh]] error bound.<ref>{{Cite journal |last=Johnsonbaugh |first=Richard |date=October 1979 |title=Summing an Alternating Series |url=https://www.jstor.org/stable/2321292 |journal=The American Mathematical Monthly |volume=86 |issue=8 |pages=637–648 |doi=10.2307/2321292|jstor=2321292 }}</ref> If one can apply the property an infinite number of times, [[Series acceleration#Euler's transform|Euler's transform]] applies.<ref>{{cite arXiv |last=Villarino |first=Mark B. |date=2015-11-27 |title=The error in an alternating series |class=math.CA |eprint=1511.08568 }}</ref>