例3将f(x)=(1+x)a∈R)展开成x的幂级数 解∵f(x)=α(αx-1)…(a-n+1)(1+x)", f(0)=a(a-1)…(a-n+1),(n=0,,2, ala 1+v+ 2 c(a-1)…(0-n+1) x+∴ x+∷ 2! ∵ liman+1|0-n h+1/, R=1 n→>oL例 3 将f (x) = (1+ x) (  R)展开成x的幂级数.  解 ( ) ( 1) ( 1)(1 ) , (n) n f x n x −  =   −   − + + (0) ( 1) ( 1), ( ) f =   −  − n + n  (n = 0,1,2,)    +   −  − + + +   − +  + n x n n x x ! ( 1) ( 1) 2! ( 1) 1 2 nn n aa 1 lim + →   + 1  − = n n = 1 ,  R = 1