Exponential function: Difference between revisions

Revision as of 22:24, 28 May 2025

Revision as of 04:24, 29 May 2025

Line 58:

\end{align}</math>

for every [[real number]] <math>x</math> and every positive real number <math>y.</math>

There is a relation between logarithm and exp(x):

:<math>\exp\left(\frac{\log(n)}{2} + \frac{1}{4} \log(\log(n))\right) = \sqrt{n} \cdot \log(n)^{1/4}</math>

===Power series===

Line 82 ⟶ 79:

''The exponential function is the [[limit (mathematics)|limit]], as the integer {{mvar|n}} goes to infinity,<ref name="Maor"/><ref name=":0" />

<math display=block>\exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n.</math>

The convergence can be improved:

<math>\exp(x)=\lim_{n \to \infty} \left(1 + \frac{x}{n + \frac{1}{6} \sin\left(\frac{\pi}{n}\right)}\right)^n</math>

By continuity of the logarithm, this can be proved by taking logarithms and proving

<math display=block>x=\lim_{n\to\infty}\ln \left(1+\frac xn\right)^n= \lim_{n\to\infty}n\ln \left(1+\frac xn\right),</math>