Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 19

Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 19

Published $\text {1968}$

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$19 \quad$ Series of Constants

Arithmetic Series

Arithmetic Series

$19.1$: Sum of Arithmetic Sequence

$19.2$: Closed Form for Triangular Numbers

$19.3$: Odd Number Theorem

Geometric Series

Geometric Series

$19.4$: Sum of Geometric Sequence: Corollary 1

$19.5$: Sum of Infinite Geometric Sequence: Corollary 2

Arithmetic-Geometric Series

Arithmetic-Geometric Series

$19.6$: Sum of Arithmetic-Geometric Sequence

$19.7$: Sum of Infinite Arithmetic-Geometric Sequence

Sums of Powers of Positive Integers

$19.8$: Sum of Powers of Positive Integers

$19.9$: Closed Form for Triangular Numbers: Proof using Bernoulli Numbers

$19.10$: Sum of Sequence of Squares: Proof using Bernoulli Numbers

$19.11$: Sum of Sequence of Cubes: Proof using Bernoulli Numbers

$19.12$: Sum of Sequence of Fourth Powers: Proof using Bernoulli Numbers

$19.13$: Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers

Series involving Reciprocals of Powers of Positive Integers

$19.14$: Newton-Mercator Series for $\ln 2$

$19.15$: Leibniz's Formula for $\pi$

$19.16$: Sum of Sequence of Reciprocals of $3 n + 1$ Alternating in Sign

$19.17$: Sum of Sequence of Reciprocals of $4 n + 1$ Alternating in Sign

$19.18$: Sum of Sequence of Reciprocals of $3 n + 2$ Alternating in Sign

$19.19$: Basel Problem

$19.20$: Riemann Zeta Function of $4$

$19.21$: Riemann Zeta Function of $6$

$19.22$: Sum of Reciprocals of Squares Alternating in Sign

$19.23$: Sum of Reciprocals of Fourth Powers Alternating in Sign

$19.24$: Sum of Reciprocals of Sixth Powers Alternating in Sign

$19.25$: Sum of Reciprocals of Squares of Odd Integers

$19.26$: Sum of Reciprocals of Fourth Powers of Odd Integers

$19.27$: Sum of Reciprocals of Sixth Powers of Odd Integers

$19.28$: Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign

$19.29$: Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign in Pairs

$19.30$: Sum of Sequence of Products of Consecutive Odd Reciprocals: Corollary

$19.31$: Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals: Corollary

$19.32$: Sum of Sequence of Products of Squares of Consecutive Odd Reciprocals

$19.33$: Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals

$19.34$: Definite Integral from $0$ to $1$ of $\dfrac {u^{a - 1} } {1 + u^d}$

$19.35$: Riemann Zeta Function at Even Integers

$19.36$: Sum of Reciprocals of Even Powers of Odd Integers

$19.37$: Sum of Reciprocals of Even Powers of Integers Alternating in Sign

$19.38$: Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign

Miscellaneous Series

$19.39$: Lagrange's Cosine Identity

$19.40$: Sine Form of Lagrange's Sine Identity

$19.41$: Sum of Infinite Series of Product of Power and Cosine

$19.42$: Sum of Infinite Series of Product of Power and Sine

$19.43$: Sum of Series of Product of Power and Cosine

$19.44$: Sum of Series of Product of Power and Sine

The Euler-Maclaurin Summation Formula

$19.45$: Euler-Maclaurin Summation Formula

The Poisson Summation Formula

$19.46$: Poisson Summation Formula

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