You can also use TaylorSeries.jl to do Taylor series manipulations:
julia> using TaylorSeries
julia> n = 16
16
julia> t = Taylor1(Rational{Int}, n)
1//1 t + 𝒪(t¹⁷)
julia> ee = sum( (-t)^i / factorial(i) for i in 0:n )
1//1 - 1//1 t + 1//2 t² - 1//6 t³ + 1//24 t⁴ - 1//120 t⁵ + 1//720 t⁶ - 1//5040 t⁷ + 1//40320 t⁸ - 1//362880 t⁹ + 1//3628800 t¹⁰ - 1//39916800 t¹¹ + 1//479001600 t¹² - 1//6227020800 t¹³ + 1//87178291200 t¹⁴ - 1//1307674368000 t¹⁵ + 1//20922789888000 t¹⁶ + 𝒪(t¹⁷)
julia> 2 * (1 - t * ee - ee) / ((1 - ee)^2)
1//1 + 1//3 t - 1//90 t³ + 1//2520 t⁵ - 1//75600 t⁷ + 1//2395008 t⁹ - 691//54486432000 t¹¹ + 1//2668723200 t¹³ - 59513//156920924160000 t¹⁵ - 42397//94152554496000 t¹⁶ + 𝒪(t¹⁷)
(Note that although exp(t) is defined for a Taylor series, it currently converts the arguments to floats.)
You can also bound the remainder using TaylorModels.jl.