Question

Let S_n = 1 · (n –1) + 2· (n – 2) + 3 · (n–3) +…. + (n –1) · 1 , n ≥ 4.

The sum \(\displaystyle\sum_{n=4}^{∞}(\frac{2S_n}{n!}-\frac{1}{(n-2)!})\) is equal to :

(1) \(\frac{e-1}{3}\) 

(2) \(\frac{e-2}{6}\)

(3) \(\frac{e}{3}\)

(4) \(\frac{e}{6}\)

Answer 1

Option : (1). \(\frac{e-1}{3}\)

Let T_r = r(n – r)

T_r = n_r – r²