MathDB

(1) For integer

n=0,\ 1,\ 2,\ \cdots

and positive number

a_{n},

let

f_{n}(x)=a_{n}(x-n)(n+1-x).

Find

a_{n}

such that the curve

y=f_{n}(x)

touches to the curve

y=e^{-x}.

(2) For

f_{n}(x)

defined in (1), denote the area of the figure bounded by

y=f_{0}(x), y=e^{-x}

and the

y

-axis by

S_{0},

for

n\geq 1,

the area of the figure bounded by

y=f_{n-1}(x),\ y=f_{n}(x)

and

y=e^{-x}

S_{n}.

Find

\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).

191