How to differentiate power series starting from 2 for e^x?

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The discussion focuses on differentiating the power series for e^x, specifically starting from the index 2 instead of 1. It is noted that starting from k = 1 results in a term that equals zero, which is why the series should begin at 2. The participants agree that the index in the sum notation needs to be adjusted accordingly. By computing the series term by term, it confirms that the expansion accurately represents e^x. This clarification helps in understanding the proper formulation of the power series.
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Homework Statement


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 + ...

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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##.
 

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