(A) e/2

(B) √e

(C) 2/e

(D) 1/e

Correct Answer: 1/e

Explanation

Consider the given series:

Σ (-1)^k+1 k(k+1)/k!

Rewrite the numerator:

k(k+1) = k² + k

So the series becomes:

Σ (-1)^k+1 (k²/k! + k/k!)

Using standard results:

Σ (-1)^k / k! = 1/e

Σ (-1)^k k / k! = -1/e

Σ (-1)^k k² / k! = 1/e

Substitute and simplify:

Required sum = 1/e

Hence, the value of the given series is 1/e.

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