Does this series that resembles the Maclaurin series of $\cosh x$ converge? Does this series converge?
$$\left(\frac{f_{i}^2}{2!} +\frac{f_{i}^4}{4!}\left(2\kappa c\right)+\frac{f_{i}^6}{6!}\left(2\kappa c\right)^2 + \frac{f_{i}^8}{8!}\left(2\kappa c\right)^3+ \cdots\right), \tag{1}$$
where $k, c \geq 0$. I am trying to establish a relation with the Maclaurin series of $\cosh x$:
$$ 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots=\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}. \tag{2}$$

Multiply the series given by Eq. (1) by $\frac{2 \kappa c}{2\kappa c}$
$$\frac{1}{2\kappa c}\left(\frac{f_{i}^2}{2!}\left(2\kappa c \right) +\frac{f_{i}^4}{4!}\left(2\kappa c\right)2+\frac{f_{i}^6}{6!}\left(2\kappa c\right)^3 + \frac{f_{i}^8}{8!}\left(2\kappa c\right)^4+ \cdots\right), $$
Add a zero to the above expression
$$\frac{1}{2\kappa c}\left(1+\frac{f_{i}^2}{2!}\left(2\kappa c \right) +\frac{f_{i}^4}{4!}\left(2\kappa c\right)2+\frac{f_{i}^6}{6!}\left(2\kappa c\right)^3 + \frac{f_{i}^8}{8!}\left(2\kappa c\right)^4+ \cdots -1\right), $$
comparing with Eq. (2) of the question, we can establish the relation with $\cosh x$
$$\frac{1}{2\kappa c}\left(\sum_{n=0}^{\infty} \frac{\left[f_{i}\left(2\kappa c\right)^{\frac{1}{2}}\right]^{2n}}{2n!}- 1\right) =\frac{\cosh \left[f_{i}\left(2\kappa c\right)^{\frac{1}{2}} \right]-1}{2\kappa c}.$$
Which is the result given by @Sangchul Lee.
