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53 .   S u p p o s e   p x   =   a 0 + a 1 x + a 2 x 2 + . . . . . . . + a n x n .   I f   p x e x - 1 - 1   f o r   a l l   x 0 . p   P r o v e   t h a t   a 1 + 2 a 2 + . . . . + n a n 1 .

pxex-1-1Put x=1p1e1-1-1p11-1p10But a0, ap1=0Let fx=ex-1-1f1=0pxex-1-1pxfxPut x=1+hp1+hf1+hp1+h-0f1+h-0p1+h-p1f1+h-f1Divide by h both sidesp1+h-p1hf1+h-f1hNow take limitlimh0p1+h-p1hlimh0f1+h-f1hlimh0p1+h-p1hlimh0f1+h-f1hBy definition, limh0p1+h-p1h=p'1 and limh0f1+h-f1h=f'1p'1f'1Now f'x=ex-1f'1=e1-1=1p'x=a1+2a2x+...+nanxn-1p'1=a1+2a2+...+nana1+2a2+...+nan1a1+2a2+...+nan1

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