Table of Integrals, Series, and Products, 8th Edition

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Introduction

0.1 Finite sums

0.11 Progressions

0.111¹² Arithmetic progression.

${\displaystyle \sum _{k=0}^{n-1}(a+kr)=\frac{n}{2}[2a+(n-1)r]=\frac{n}{2}(a+1)=\frac{1}{2r}(a+1)(l-a+r)}$

  [l=a + (n-1)r is the last term]

0.112 Geometric progression.

${\displaystyle \sum _{k=1}^{n}a{q}^{k-1}=\frac{a\left(q^n-1\right)}{q-1}}$

  [q ≠ 1]

0.113 Arithmetic-geometric progression.

${\displaystyle \sum _{k=0}^n\left(a+kr\right)q^k=\frac{a-[a+(n-1)r]q^n}{1-q}+\frac{rq(1-q^{n-1})}{{(1-q)}^2}}$

  [q ≠ 1, n > 1] JO (5)

0.114⁸

${\displaystyle \sum _{k=1}^{n-1}{k}^2{x}^k=\frac{(-n^2+2n-1)x^{n+2}+(2n^2-2n-1)x^{n+1}-n^{2+1}-n^2{x}^n+x^2+x}{{(1-x)}^3}}$

0.12 Sums of powers of natural ...