63. (Newton)设$x$为整数并且$0leq xleq n$，试证
$$f(x)=f(0)+inom{x}{1}Delta f(0)+inom{x}{2}Delta^{2}f(0)+cdots+inom{x}{n}Delta^{n}f(0)$$
64.(牛顿-格雷戈里的插值公式) 试证对任意$x$, 有
$$f(x)=f(0)+inom{x}{1}Delta f(0)+inom{x}{2}Delta^{2}f(0)+cdots+inom{x}{n}Delta^{n}f(0)+R_{n}(x)$$
此地余项$R_{n}(x)=frac{P_{n}(x)}{Q_{n}(x)}$由下式决定:
egin{equation*}
P_{n}(x)=left|
egin{array}{ccccc}
1 & 0 & cdots&0&f(0)\
1 & 1^{1} & cdots&1^{n}&f(1)\
cdots&cdots&cdots&cdots&cdots\
1 & n^{1} & cdots&n^{n}&f(n)\
1 & x^{1} & cdots&x^{n}&f(x)
end{array}
ight|
end{equation*}
egin{equation*}
Q_{n}(x)=left| egin{array}{cccc}
1 & 1^{2} & cdots&1^{n}\
2 & 1^{2} & cdots&2^{n}\
cdots&cdots&cdots&cdots\
n& n^{2} & cdots&n^{n}
end{array}
ight|
end{equation*}
74. (贝努利求和公式) 设$f(x)$为任意对$x=1,2,cdots,n$有定义的函数,则
$$sum_{k=1}^{n}f(k)=inom{n}{1}f(1)+inom{n}{2}Delta f(1)+cdotsinom{n}{k}Delta^{k-1}f(1)+cdots+inom{n}{n}Delta^{n-1}f(1)$$
79. (欧拉转换公式) 设级数$sum (-1)^{n-1}f(n)$收敛(不必为交错级数), 则对任意非负整数$p$, 有
$$sum_{n=1}^{infty}(-1)^{n-1}f(n)=frac{1}{2}f(1)-frac{1}{2^{2}}Delta f(1)+frac{1}{2^{3}}Delta^{2}f(1)+cdots+frac{1}{2^{k+1}}Delta^{k}f(1)+cdots$$