for d/dx(e^x) , the series should be start from 2 rather than 1 , right ? because when k = 1 , the circled part would become 0 , the series for (e^x) is 1 + x + ...
The circled part is correct. There, the index of the sum notation is just shifted up by one, therefore the indices appearing in the term must be shifted down by one. If you compute the series terms by terms you should find that it's indeed the power expansion of ##e^x##.
