The alternating collection take a look at is a part of the usual calculus curriculum. It says that when you truncate an alternating collection, the rest is bounded by the primary time period that was omitted. This truth goes by in a blur for many college students, but it surely turns into helpful later if you want to do numerical computing.
To be extra exact, assume we’ve a collection of the shape
the place the ai are optimistic and monotonically converge to zero. Then the tail of the collection is bounded by its first time period:
The extra we will say concerning the conduct of the ai the extra we will say concerning the the rest. To date we’ve assumed that these phrases go monotonically to zero. If their variations
additionally go monotonically to zero, then we’ve an higher and decrease sure on the truncation error:
If the variations of the variations,
additionally converge monotonically to zero, we will get a bigger decrease sure and a smaller higher sure on the rest. Generally, if the variations as much as order ok of the ai go to zero monotonically, then the rest time period may be bounded as follows.
Supply: Mark B. Villarino. The Error in an Alternating Collection. American Mathematical Month-to-month, April 2018, pp. 360–364.
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