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Number Theory Level 3

Which option is the closest to the value of the summation below?

∑ n = 1 1000 ln ⁡ ( n n ) \large \sum_{n=1}^{1000}\ln(n^n)

Notation: ln ⁡ ( ⋅ ) \ln(\cdot) denotes the natural logarithm function .

3.2 × 1 0 6 3.2 \times 10^6 3.8 × 1 0 5 3.8 \times 10^5 3.8 × 1 0 6 3.8 \times 10^6 3.2 × 1 0 5 3.2 \times 10^5

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1 solution

∑ n = 1 1000 ln ⁡ ( n n ) = ∑ n = 1 1000 n ln ⁡ n ≈ ∫ 1 1000 x ln ⁡ x d x By integration by parts = x 2 ln ⁡ x 2 ∣ 1 1000 − ∫ 0 1000 x 2 d x = 500 000 ln ⁡ 1000 − x 2 4 ∣ 1 1000 = 500 000 ln ⁡ 1000 − 250 000 + 1 4 ≈ 3.204 × 1 0 6 \begin{aligned} \sum_{n=1}^{1000} \ln (n^n) & = \sum_{n=1}^{1000} n \ln n \\ & \approx \int_1^{1000} x \ln x \ dx & \small \color{#3D99F6} \text{By integration by parts} \\ & = \frac {x^2 \ln x}2\ \bigg|_1^{1000} - \int_0^{1000} \frac x2 \ dx \\ & = 500\ 000 \ln 1000 - \frac {x^2}4 \ \bigg|_1^{1000} \\ & = 500\ 000 \ln 1000 - 250\ 000 + \frac 14 \\ & \approx 3.204 \times 10^6 \end{aligned}

Therefore, the closest value is 3.2 × 1 0 6 \boxed{3.2 \times 10^6} .

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