example of summation by parts

PropositionPlanetmathPlanetmath. The series n=1sinnφn and n=1cosnφn convergePlanetmathPlanetmath for every complex value φ which is not an even multiple of π.

Proof. Let ε be an arbitrary positive number. One uses the

sinφ+sin2φ++sinnφ=sin(n+12)φ-sinφ22sinφ2, (1)
cosφ+cos2φ++cosnφ=-cos(n+12)φ+cosφ22sinφ2, (2)

proved in the entry “example of telescoping sum (http://planetmath.org/ExampleOfTelescopingSum)”. These give the


for any  n=1, 2, 3,. We want to apply to the series n=1cosnφn the Cauchy general convergence criterion (http://planetmath.org/CauchyCriterionForConvergence) for series. Let us use here the short notation

cosNφ+cos(N+1)φ++cos(N+p)φ:=SN,N+p(p=0, 1, 2,).

Then, utilizing Abel’s summation by partsPlanetmathPlanetmath, we obtain


the last form is gotten by telescoping (http://planetmath.org/TelescopingSum) the preceding sum and before that by using the identityPlanetmathPlanetmathPlanetmath


Thus we see that


for all  natural numbersMathworldPlanetmath P as soon as  N>2Kφε. According to the Cauchy criterion, the latter series is convergent for the mentioned values of φ. The former series is handled similarly.

Title example of summation by parts
Canonical name ExampleOfSummationByParts
Date of creation 2013-03-22 17:27:56
Last modified on 2013-03-22 17:27:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 40A05
Related topic ExampleOfTelescopingSum
Related topic SineIntegralInInfinity
Related topic ExampleOfSolvingTheHeatEquation