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Question Number 130733 by bemath last updated on 28/Jan/21

Commented by EDWIN88 last updated on 28/Jan/21

$${10}^{e-1}$$

Answered by Dwaipayan Shikari last updated on 28/Jan/21

$$10\sqrt{10\sqrt[3]{10\sqrt[4]{10\sqrt[5]{10..}}}}$$$$={10}^{1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+....}={10}^{e-1}$$

Commented by bemath last updated on 28/Jan/21

$$1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+...=\mathrm{e}-1?$$

Commented by Dwaipayan Shikari last updated on 28/Jan/21

$$yes$$

Commented by bemath last updated on 28/Jan/21

$${\mathrm{e}}^{\mathrm{x}}=1+\mathrm{x}+\frac{{\mathrm{x}}^2}{2!}+\frac{{\mathrm{x}}^3}{3!}+\frac{{\mathrm{x}}^4}{4!}+...$$$${\mathrm{e}}^1=1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+...$$$$\mathrm{yes}...$$

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