New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE Below is the OCR-scanned text from this page: 3. Elementary Analytical Methods 3.1. Binomial Theorem and Binomial Coeffi- cients; Arithmetic and Geometric Progres- sions; Arithmetic, Geometric, Harmonic and Generalized Means Binomial Theorem 3.1.1 (a + b) " = an + c) an- 1 b + C) a" - 2 b 2 +G)an-3b3+ . . . +b" (n a positive integer) Binomial Coefficients (see chapter 24) 3.1.2 * C ) = n c l k = k! (n- k) ! k! n(n-1) . . . (n-k+l)- k-n-1 n! - 3*1*3 c)=(nTk)=(-l)t( k ) n+1 3.1.4 ( k )=C)+(kTl) 3.1.5 C)=C)=l 3.1.6 3.1.7 l - c ) + c ) - . . . +(-1)"C)=O l+c)+C)+. . . +(")=2" n Table of Binomial Coefficients 3.1.8 For a more extensive table see chapter 24. 3.1.9 Sum of Arithmetic Progression to n Terms a-t-b+d)+(a+2d)+ * . . +(a+(n--l)d) =w~+~n(n-l)d=- 1 n ( a f l ) , 2 last term in series=l=a+(n-l)d Sum of Geometric Progression to n Terms 3.1.10 sn=a+ar+ar2+ . . . + u r n - l = ' H 1-r lim s,=a/(l-r) (-lO, k=l, 2, . . ., n) Harmonic Mean of n Quantities H 3.1.13 -=- 1 1 1 1 (-+-+ . . . +') (ak>O, k=l, 2, . . ., n) H n al a2 a n Generalized Mean 3.1.14 3.1.15 M(t)=O(t H , equality if and only if al=az= . . . =an 3.2.2 min. a