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Question Number 123672 by mathmax by abdo last updated on 27/Nov/20

$$\mathrm{find}\mathrm{nature}\mathrm{of}\mathrm{the}\mathrm{serie}\sum _{\mathrm{n}=0}^{\mathrm{\infty }}\cos \left(\pi \sqrt{{\mathrm{n}}^2+\mathrm{n}+1}\right)$$

Answered by mnjuly1970 last updated on 27/Nov/20

$$\sqrt{n^2+n+1}\stackrel{n\to \mathrm{\infty }}{\approx }(n+\frac{1}{2})$$$$\sum _{n⩾0}cos\left(\sqrt{n^2+n+1}\right)\approx \mathrm{\Sigma }-sin\left(n\pi \right)$$

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