Exercise 2.7.12

[Summation-by-parts] Let ( x n ) and ( y n ) be sequences, let s n = x 1 + x 2 + ⋯ + x n and set s 0 = 0 . Use the observation that x j = s j − s j − 1 to verify the formula

∑ j = m n x j y j = s n y n + 1 − s m − 1 y m + ∑ j = m n s j ( y j − y j + 1 )

Answers

Since x j = s j − s j − 1 we can rewrite the sum as

∑ j = m n x j y j = ∑ j = m n y j ( s j − s j − 1 ) = s n y n + 1 − s m y m − 1 + ∑ j = m n s j ( y j − y j + 1 )

In the last part we combine each s j y j term with the − s j y j + 1 term which is next in the sum, then we add some correction terms for the start and ending points.

Note the symmetry here, we can turn a sum ∑ ⁡ y j ( s j − s j − 1 ) into a sum ∑ ⁡ s j ( y j − y j + 1 ) (at the cost of some correction terms). This is a useful pattern to keep in mind.

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2022-01-27 00:00
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