On the Riemann-Zeta Function!

Calculus Level 5

f(n)=(ζ(2)+ζ(3)++ζ(n+1)n)nα\large{f(n) = \left( \dfrac{\zeta(2) + \zeta(3) + \cdots + \zeta(n+1)}{n} \right)^{n^\alpha}}

Let f(n)f(n) be a function defined as above, where ζ(k)=p=11pk\zeta(k) = \displaystyle \sum_{p=1}^\infty \dfrac{1}{p^k}.
Let A=limnf(n)A = \displaystyle \lim_{n \to \infty} f(n), when α=1\alpha = 1.
Let B=limnf(n)B = \displaystyle \lim_{n \to \infty} f(n), when α=12\alpha = \dfrac12.

Find the value of A+BA+B up to three decimal places.

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