How do you find the power series for #f'(x)# and #int f(t)dt# from [0,x] given the function #f(x)=Sigma x^n/lnn# from #n=[2,oo)#?

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Answer 1 · 2017-03-23T12:03:15

See below

For:

#f(x)=sum_(n = 2)^(oo) x^n/lnn#

We differentiate wrt x:

#f'(x)= sum_(n = 2)^(oo) (n x^(n-1))/lnn#

And:

#int_0^x f(t) \ dt = int_0^x sum_(n = 2)^(oo) t^n/lnn \ dt#

#= sum_(n = 2)^(oo) (x^(n+1))/((n+1) lnn )#