q149

The value of Σ (−1)^(k+1) k(k+1)/k! is

Q.

∑_k=1^∞ (−1)^k+1 k(k+1) / k!

is

1. e / 2

2. 2 / e

3. √e

4. 1 / e

Correct Answer: 2 / e

Explanation

Step 1:

Given series is: ∑_k=1^∞ (−1)^k+1 [ k(k+1) / k! ]

Step 2:

Expand the numerator: k(k+1) = k² + k

Step 3:

Split the term:
(k² + k)/k! = k/(k−1)! + 1/(k−1)!

Step 4:

So the series becomes sum of two standard exponential series:
∑ (−1)^k+1 / (k−1)! + ∑ (−1)^k+1 k/(k−1)!

Step 5:

Using known results of exponential series and simplifying, the value of the given series is:

2 / e

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