Which option is the closest to the value of the summation below?
Notation: denotes the natural logarithm function .
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n = 1 ∑ 1 0 0 0 ln ( n n ) = n = 1 ∑ 1 0 0 0 n ln n ≈ ∫ 1 1 0 0 0 x ln x d x = 2 x 2 ln x ∣ ∣ ∣ ∣ 1 1 0 0 0 − ∫ 0 1 0 0 0 2 x d x = 5 0 0 0 0 0 ln 1 0 0 0 − 4 x 2 ∣ ∣ ∣ ∣ 1 1 0 0 0 = 5 0 0 0 0 0 ln 1 0 0 0 − 2 5 0 0 0 0 + 4 1 ≈ 3 . 2 0 4 × 1 0 6 By integration by parts
Therefore, the closest value is 3 . 2 × 1 0 6 .