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Question Number 77556 by behi83417@gmail.com last updated on 07/Jan/20

$$1.1\times 1!\times 2!+2\times 2!\times 3!+3\times 3!\times 4!+....=?$$$$2.\frac{1+\sqrt{2}}{2+\sqrt{3}}+\frac{2+\sqrt{3}}{3+\sqrt{4}}+\frac{3+\sqrt{4}}{4+\sqrt{5}}+......=?$$$$3.\frac{1}{1+\sqrt{2}+\sqrt{3}}+\frac{2}{\sqrt{2}+\sqrt{3}+\sqrt{4}}+\frac{3}{\sqrt{3}+\sqrt{4}+\sqrt{5}}+....=?$$

Commented by mr W last updated on 07/Jan/20

$$1.\to \mathrm{\infty },clear$$$$2.a_n\to \frac{n}{n}\to 1,\mathrm{\Sigma }\to \mathrm{\infty }$$$$3.a_n\to \frac{\sqrt{n}}{3}\to \mathrm{\infty },\mathrm{\Sigma }\to \mathrm{\infty }$$

Commented by behi83417@gmail.com last updated on 08/Jan/20

$$\mathrm{thak}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{dear}\mathrm{master}.$$

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