How can $e^x$ be restated for small $x$?

Question

Suppose I have the following equation:

\begin{equation} 1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau. \end{equation}

If I make two assumptions

1. g is a periodic function with period $T$ and mean zero.

2. $\gamma$ is small

somehow my equation changes to this:

$$ 1=S_0T_n+\lambda\int^{t_{n+1}}_{t_n}g(\tau)d\tau-\lambda \gamma \int^{t_{n+1}}_{t_n}(t_{n+1}-\tau)g(\tau)d\tau. $$

How is this restatement of the equation achieved?

Answer 1

This answer is thanks to @MPW.

The restatement depends on the approximation

$$ e^x\approx x+1, $$

which holds for small $x$. Here, the relevant small variable is $\gamma$.