The value of k=1 (-1)k+1 ( k(k+1)k! ) is
(4) 1/e
k(k+1)k! = k+1(k-1)!
Put m = k − 1 (m = 0, 1, 2, …): (-1)k+1 = (-1)m
S = m=0 (-1)m m + 2m!
= m=1 (-1)m(m-1)! + 2 m=0 (-1)mm!
= - j=0 (-1)jj! + 2e-1
= -e-1 + 2e-1
S = 1e
Answer: (4)
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