Sunday, 18 November 2018

calculus - General Cesaro summation with weight




Assume that an is a convergent sequence of complex numbers and {λn} is a sequence of positive real numbers such that k=0λk=





Then, show that,
limn1nk=0λknk=0λkak==limnan



(Note that : This is more general than the special case where, λn=1)



Answer



Let ε>0 and Nsuch that |akl|ε for all k>N
Then, for n>N we have,

|nk=0λkaknk=0λkl|=|nk=0λk(akl)nk=0λk|=|Nk=0λk(akl)+nk=Nλk(akl)nk=0λk|Mnk=0λk+nk=Nλk|akl|εnk=0λkMnk=0λk+ε0


since Nk=0λk.
Where M=|Nk=0λk(akl)|


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