Solution of an equation with polynomial and exponential terms

Question

Can anyone solve for $t$ the equation: $$ e^t=\frac{1-nt}{1-t} $$

with $n \in \mathbb N$ (known) and $t>0$. Online solvers give an answer only for specific values of $n$, but I need a general formula for $t$.

Thanks.

Answer 1

In this special case, you can guess t = 0 as a solution.

This solution is, moreover, independent from n.

Perhaps the solution can be found by expanding the right side of the equation. It is a geometric series (up to a constant).

The expansion of the right side is

$$1 + (1-n) (t+t^2+t^3+t^4+...)$$