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

### Miscellaneous Series

$19.39$: Sum of Cosines of Multiples of Angle
$19.40$: Sum of Sines of Multiples of Angle
$19.41$: 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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