As a follow up of my previous question here is an improved version of the series: <munderover>

Alaina Marshall

Alaina Marshall

Answered question

2022-05-19

As a follow up of my previous question here is an improved version of the series:
i = 1 2 N R i ( 1 1 p ) i ( 1 2 R ( 2 N R i ) Δ 2 R )

Answer & Explanation

Louis Lawrence

Louis Lawrence

Beginner2022-05-20Added 10 answers

Note that a little algebra will transform your sum into
i = 1 2 N R ( K i + Δ 2 R i 2 ) ( 1 1 p ) i ,
where K is an easily computed constant.
The answer to the previous post tells you how to find 1 2 N R i x i . Multiply that sum by K. The only remaining task is to find 1 2 N R i 2 x i
This is done using more or less the same idea as the preceding answer by Marvis. Let f ( x ) = 1 + x + x 2 + + x n . There is an explicit closed form for f ( x ). Note that
x f ( x ) = i = 0 n i x i .
If g ( x ) = x f ( x ), then by the same trick
x g ( x ) = i = 0 n i 2 x i .

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