MHB Finding the function of a maclaurin series

[tmt1]

I need to find the function for this Maclaurin series

$$1 - \frac{5^3x^3}{3!} + \frac{5^5x^5}{5!} - \frac{5^7x^7}{7!} ...$$

I can derive this sigma:

$$1 + \sum_{n = 2}^{\infty} \frac{(-1)^{n - 1} 5^{2n - 1} x^{2n - 1}}{(2n - 1)!}$$

But I'm not sure how to get this function from this series.

 

[Greg]

The sum may be written as

$$1+\sum_{n=1}^\infty\dfrac{(-1)^n}{(2n+1)!}(5x)^{2n+1}$$

which is equivalent to $\sin(5x)-5x+1$.

See here for a summary of some well-known MacLaurin series.