Verifique que \(sinh \space(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}\)
R:
Primeiramente, partindo da definição de seno hiperbólico:
\( sinh \space(x) = \frac{e^x - e^{-x}}{2} \)
\( = \frac{\sum_{n=0}^{\infty} \frac{x^n}{n!}}{2} - \frac{\sum_{n=0}^{\infty} \frac{(-x)^n}{n!}}{2} \)
\( = \frac{1}{2} \cdot (\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} + \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} - \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} + \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} )\)
\( = \frac{1}{2} \cdot (2 \cdot \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} )\)
\( = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \blacksquare \)