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here is my table
n f^n x f^n 0
0 coshx 1
1 sinhx 0
2 coshx 1
3 sinhx 0
4 coshx 1

these are even values of the exponent, so we use the other formula with the 2n

ooops

\[\sum_{n=0}^{\infty} \frac{...x^{(2n)}}{(2n)!}\]

so just take out the dots lol

oooh because their just zero's and one's ?

yup
try out some values of n
plug in
n=0
n=1
n=2
etc.
you will see you get your series

check out my table

you do not need to represent the 0 terms

you only need to represent the resulting series

\[f(x)=cosh(x)=1+\frac{x^2}{2!}+\frac{x^4}{4!}+....\]
?

that agrees with your results doesn't it?

makes sense

welcome!