Sunday, January 14, 2018

sequences and series - infinite summation of exponential functions




I've discovered through Wolfram Alpha that



t=1ebt=1eb1



What are the steps of derivation here? According to infinite summation of power series:



t=1pt=11p,



I expected the solution to be




t=1(eb)t=11eb.



What am I getting wrong?



In extension, how do I derive



t=1eb(t1) ?


Answer



Your second formula isn't quite right: if |p|<1, then

t=1pt=p1p


Using this with p=eb yields
t=1ebt=eb1eb=1eb1

as claimed.


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