1

Elementary Functions

1.1 Power of Binomials

1.11 Power series

1.110

(1+x)q=1+qx+q(q1)2!x2++q(q1)(qk+1)k!xk+=k=0(qk)?xk

si1_e

If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If q = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. FI II 425

1.111

(a+x)n=k=0n(nk)xkank

1.112

1. 

(1+x)1=1x+x2x3+=k=1(1)k1xk1

(see also 1.121 2)

2. 

(1+x)2=1

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