i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am pretty sure about the calculations but then again, if someone smells something fishy, feel free, i will gladly provide the details. Below are the resultant series:
After summing up the terms i get something which looks like a power series but whose convergence am not very familiar with. the terms added are:
$f0=2*(1-e)e^x$
$f1=e^{-x}+e^{-x+1}+e^{-x+2}-e^{-x+3}$
$f2=e^{-x}-2*e^{-x+1}+2*e^{-x+3}-e^{-x+4}$
... After obtaining more of such terms and adding the first 8
$2*(1-e)e^x - 4e^{-x+1}+16e^{-x+2}-30 e^{-x+3}+30e^{-x+4}$ $+-6e^{-x+5}-24e^{-x+6}+33 e^{-x+7}-21e^{-x+8}+$ $7e^{-x+9}-e^{-x+10}$
Is it safe for me to assume that as the number of terms grows the power of the exponential function also grows and as it is negative, the terms eventually become zero. So it becomes zero the number of functions goes towards infinity?
Thanks alot