For \(x\gt0\), the power series \(\displaystyle 1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+...+ (-1)^n \frac{x^{2n}}{(2n+1)!}+...\) converges to which of the following?

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Note that only the even powers are present and the coefficients alternate. This implies a sine or cosine function.

Normally, the expansion of cosine is:

$$ 1-\frac{x^2}{2!}+\frac{x^4}{4!} + ... $$

The expansion of sine is:

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

Neither of these fit, but if we divide each term of the Taylor expansion of sine by \(x\), we obtain the give series.

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