Leibniz’ estimate for alternating series
Theorem (Leibniz 1682). If and , then the alternating series
Proof. The convergence of (1) is proved here (http://planetmath.org/ProofOfAlternatingSeriesTest). Now denote the sum of the series by and the partial sums of it by . Suppose that (1) is truncated after a negative . Then the remainder term
may be written in the form
The former shows that is positive as the first omitted and the latter that . Similarly one can see the assertions true when the series (1) is truncated after a positive .
A pictorial proof.
As seen in this diagram, whenever , we have
. Thus the partial sums form a Cauchy sequence, and hence converge. The limit lies in the of the spiral, strictly in and for any . So the remainder after the th must have the same direction as and lesser magnitude.
Example 1. The alternating series
does not fulfil the requirements of the theorem and is divergent.
Example 2. The alternating series
satisfies all conditions of the theorem and is convergent.
|Title||Leibniz’ estimate for alternating series|
|Date of creation||2014-07-22 15:34:38|
|Last modified on||2014-07-22 15:34:38|
|Last modified by||pahio (2872)|
|Synonym||Leibniz’ estimate for remainder term|