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can't figure out infinite sum of integral, halfway there

Hi!

please could someone point out where i went wrong. the question is attached (sorry for the lack of latex)

i need to find

lim(as a tends to inf) of SUM (from n=1 to inf) of INTEGRAL (from 1 to inf) of the function exp(-2nx)*(1+ x/a)^(an) dx

so far i took the limit inside the bracket to get

lim (1+x/a)^na = e^(xn)

then you get

SUM [ INTEGRAL (exp(-2nx)*exp(nx)) dx]

= SUM [INTEGRAL exp(-nx) dx]

= SUM [ -1/n *exp(-xn) ] from 1 to inf

= SUM [ -1/n *exp (-n) - 0]

and that sum is pretty funky, i don't know how to arrive at the given result :(

Re: can't figure out infinite sum of integral, halfway there

Hey cassius2020.

Hint: try looking at the Taylor series for the logarithmic function:

Taylor Series Expansions of Logarimathic Functions