MathDB

For positive integers

n, p

with

n \geq p

, define real number

K_{n, p}

as follows:

K_{n, 0} = \frac{1}{n+1}

and

K_{n, p} = K_{n-1, p-1} -K_{n, p-1}

for

1 \leq p \leq n.

(i) Define

S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots

. Find

\lim_{n \to \infty} S_n.

(ii) Find

T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots

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