I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function.
$\sigma = 1 + \int\nolimits_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 \int_{t_0}^{t_1}\mathrm dt_2 \,\varepsilon(t_1)\, \varepsilon(t_2) + \int_{t_0}^t\mathrm dt_1 \int_{t_0}^{t_1}\mathrm dt_2 \int_{t_0}^{t_2}\mathrm dt_3\, \varepsilon(t_1)\, \varepsilon(t_2)\, \varepsilon(t_3) + \cdots \;.$
The result should be
$\sigma = \exp\left(\int_{t_0}^t\mathrm dt_1\, \varepsilon(t_1)\right) = \sum_{i=0}^\infty \frac1{i!} \left(\int_{t_0}^t\mathrm dt_1 \,\varepsilon(t_1)\right)^i \;.$
However, comparing the series term by term I already fail to prove the equivalence for the third term. Can someone clear this up for me? Can the series be rewritten to an exponential after all? I recall from my quantum mechanics course that if $\varepsilon(t)$ was an operator, non-commuting with itself for different times, then the formal result would be
$\sigma = T_c \exp\left(\int_{t_0}^t\mathrm dt_1\, \varepsilon(t_1)\right) \;,$
with $T_c$ being the usual time-ordering operator. However, as I said in my case $\varepsilon(t)$ is a plain function in the real domain.