I was wondering if there is a way to make an infinite series of odd functions equal to an even function. For example, I would like to know if the next equation is valid
$$\cos(x)=\beta_1x+\beta_2x^3+\beta_3x^5+\beta_4x^7+\cdots$$
Is there any proof that says that is possible?
On
If you want to represent $\cos x$ with high accuracy with an odd powered series in a domain interval that is either strictly positive or strictly negative, see the answer to Approximating a continuous function on a positive interval with a series with arbitrary prescribed powers? . I think this is about the best that you can do. As other answers/comments alluded, if you have an odd powered series and $0$ is in the domain interval for which you want to approximate, then you can't get a good approximation everywhere in the interval because for your power series you must have $f(0) = 0$ and also $f(-x) = -f(x)$ even though $\cos 0 = 1$ and $\cos (-x) = \cos x$.
As Daniel Fischer says, any linear combination of odd functions is still odd. So if $\cos (2\pi )=1$ is approximated well, your approximation will satisfy $\cos (-2\pi)\approx -1$, which is rather far from $\cos (-2\pi )=1$ This does not strike me as a good approximation.