I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, then if there is an upper limit, $M$, to the individual claim amount, we have $$e^{Rx} \leq \frac x M e^{RM} + 1 - \frac x M\ \forall\ 0 \leq x \leq M.$$
The proof given is as follows:
$$\begin{aligned} \frac x M e^{RM} + 1 - \frac x M & = \frac x M \sum^{\infty}_{j = 0} \frac {{RM}^j} {j!} + 1 - \frac x M\\ & = 1 + \sum^{\infty}_{j = 0} \frac {R^j M^{j - 1}x} {j!}\\ & \geq 1 + \sum^{\infty}_{j = 0} \frac {{Rx}^j} {j!}\\ & = e^{Rx}, \end{aligned}$$
where the inequality is true because $$x^j \leq M^{j - 1}x\ \forall\ 0 \leq x \leq M.$$
Now, I am having issues understanding how my professor went from the first equality to the second one, that is, how he went from $$\frac x M \sum^{\infty}_{j = 0} \frac {{RM}^j} {j!} + 1 - \frac x M$$ to $$1 + \sum^{\infty}_{j = 0} \frac {R^j M^{j - 1}x} {j!}.$$ I wondered whether this was an error, in the sense that he should have gone from $$\frac x M \sum^{\infty}_{j = 0} \frac {{RM}^j} {j!} + 1 - \frac x M$$ to $$1 + \sum^{\infty}_{j = 0} \frac {R^j M^{j - 1}x} {j!} - \frac x M,$$ but he mentioned that he skipped a few steps to get from the first equality to the second one, implicitly implying that the derivation is correct.
Thus, is anyone able to shed some light on how we can get from the first equality to the second one? Any intuitive explanations or suggestions will be greatly appreciated :)
I think he may have forgotten to update the index on the sum;
$${x\over M}\sum_{j=0}^{\infty}{(RM)^j \over j!}+1-{x\over M}$$
$$={x\over M}\left({\sum_{j=0}^{\infty}{(RM)^j \over j!}}-1\right )+1$$
$$={x\over M} \sum_{j=1}^{\infty}{(RM)^j \over j!}+1$$
$$=1+\sum_{j=1}^{\infty}{R^jM^{(j-1)}x \over j!}$$