Proving $(2n-1)^n + (2n)^n ≈ (2n+1)^n$

limits proof-verification

$$\left(\dfrac{2n-1}{2n+1} \right)^n = \left(1 -\dfrac2{2n+1} \right)^n \sim \dfrac1e - \dfrac1{en^2} + \mathcal{O}(1/n^4)$$

$$\left(\dfrac{2n}{2n+1} \right)^n = \left(1 -\dfrac1{2n+1} \right)^n \sim \dfrac1{\sqrt{e}}$$

Hence, $$\left(\dfrac{2n-1}{2n+1} \right)^n + \left(\dfrac{2n}{2n+1} \right)^n \sim \dfrac1e + \dfrac1{\sqrt{e}} \approx 0.97441$$

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