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Calculus Level 4

∑ n = 1 ∞ n + 1 n 3 = C \sum_{n=1}^{\infty} \frac{n+1}{n^{3}} = C

C C can be written in the format below.

π 2 + 6 ζ ( a ) b \frac{\pi^{2} + 6\zeta(a)}{b}

Submit your answer as, a + b a+b

Details and Assumptions:

  • ζ ( x ) \zeta(x) denotes Riemann Zeta Function.


The answer is 9.

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1 solution

Chew-Seong Cheong
Apr 20, 2016

C = ∑ n = 1 ∞ n + 1 n 3 = ∑ n = 1 ∞ 1 n 2 + ∑ n = 1 ∞ 1 n 3 = ζ ( 2 ) + ζ ( 3 ) = π 2 6 + ζ ( 3 ) = π 2 + 6 ζ ( 3 ) 6 \begin{aligned} C & = \sum_{n=1}^\infty \frac{n+1}{n^3} \\ & = \sum_{n=1}^\infty \frac{1}{n^2} + \sum_{n=1}^\infty \frac{1}{n^3} \\ & = \zeta(2) + \zeta(3) \\ & = \frac{\pi^2}{6} + \zeta(3) \\ & = \frac{\pi^2+6\zeta(3)}{6} \end{aligned}

⇒ a + b = 3 + 6 = 9 \Rightarrow a + b = 3 + 6 = \boxed{9}

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Same done.

Aditya Narayan Sharma - 5 years, 1 month ago

Did the same

Aditya Kumar - 5 years, 1 month ago

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