63. (Newton)设$x$为整数并且$0\leq x\leq n$，试证
$$f(x)=f(0)+\binom{x}{1}\Delta f(0)+\binom{x}{2}\Delta^{2}f(0)+\cdots+\binom{x}{n}\Delta^{n}f(0)$$
64.(牛顿-格雷戈里的插值公式) 试证对任意$x$, 有
$$f(x)=f(0)+\binom{x}{1}\Delta f(0)+\binom{x}{2}\Delta^{2}f(0)+\cdots+\binom{x}{n}\Delta^{n}f(0)+R_{n}(x)$$
此地余项$R_{n}(x)=\frac{P_{n}(x)}{Q_{n}(x)}$由下式决定:
\begin{equation*}
P_{n}(x)=\left|
\begin{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}\right|
\end{equation*}
\begin{equation*}
Q_{n}(x)=\left| \begin{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}\right|
\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
$$\sum_{k=1}^{n}f(k)=\binom{n}{1}f(1)+\binom{n}{2}\Delta f(1)+\cdots\binom{n}{k}\Delta^{k-1}f(1)+\cdots+\binom{n}{n}\Delta^{n-1}f(1)$$
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
$$\sum_{n=1}^{\infty}(-1)^{n-1}f(n)=\frac{1}{2}f(1)-\frac{1}{2^{2}}\Delta f(1)+\frac{1}{2^{3}}\Delta^{2}f(1)+\cdots+\frac{1}{2^{k+1}}\Delta^{k}f(1)+\cdots$$