The_Cambridge_Handbook_of_Physics_Formulas

2.3 Series, summations, and progressions
27
2.3 Series, summations, and progressions
Progressions and summations
Arithmetic
progression
S n = a+(a+d)+(a+2d)+···
n number of terms
+[a+(n−1)d] (2.102) S n sum of n successive
= n terms
[2a+(n−1)d] (2.103) a first term
2
= n 2 (a+l) (2.104) d common difference
l last term
2
Geometric
progression
Arithmetic
mean
Geometric
mean
S n = a+ar+ar 2 +···+ar n−1 (2.105)
= a 1−rn
1−r
(2.106)
S ∞ =
a
1−r
(|r| < 1) (2.107)
r
common ratio
〈x〉 a = 1 n (x 1 +x 2 +···+x n ) (2.108) 〈.〉 a arithmetic mean
〈x〉 g =(x 1 x 2 x 3 ...x n ) 1/n (2.109) 〈.〉 g geometric mean
Harmonic mean 〈x〉 h = n( 1
x 1
+ 1 x 2
+···+ 1 x n
) −1
(2.110) 〈.〉 h harmonic mean
Relative mean
magnitudes
Summation
formulas
Euler’s
constant a
a γ ≃ 0.577215664...
〈x〉 a ≥〈x〉 g ≥〈x〉 h if x i > 0foralli (2.111)
n∑
i=1
n∑
i=1
n∑
i=1
n∑
i=1
i = n (n+1) (2.112)
2
i 2 = n (n+1)(2n+1) (2.113)
6
i 3 = n2
4 (n+1)2 (2.114)
i 4 = n
30 (n+1)(2n+1)(3n2 +3n−1) (2.115)
∞∑ (−1) i+1
i
∞∑
i=1
i=1
∞∑
i=1
=1− 1 2 + 1 3 − 1 +...= ln2 (2.116)
4
(−1) i+1
2i−1 =1− 1 3 + 1 5 − 1 7 +...= π 4
1
i 2 =1+1 4 + 1 9 + 1 16
+...=
π2
6
(
γ = lim 1+ 1
n→∞ 2 + 1 3 +···+ 1 )
n −lnn
(2.117)
(2.118)
i
dummy integer
(2.119) γ Euler’s constant