 Shifted integral representation of a p-series tail

ID: shifted-integral-representation-of-a-p-series-tail

N > 1

and a real exponent

p > 1

, and put

S_{p} = \sum_{n = N}^{\infty} n^{- p}

. There is a unique shift

k_{p} < N

such that

\int_{N}^{\infty} (x - k_{p})^{- p} d x = S_{p} .

Integration evaluates the left side as

(N - k_{p})^{1 - p} / (p - 1)

, yielding the displayed formula. Comparing the decreasing function

x^{- p}

with unit rectangles proves

\frac{N ^{1 - p}}{p - 1} < S_{p} < \frac{( N - 1 ) ^{1 - p}}{p - 1} .

0 < k_{p} < 1

. This is an exact reparametrization of a tail, not a claim that its shift is independent of

p

; interpolating shifts for different exponents is an additional approximation.

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