How do you find the power series for f'(x) and int f(t)dt from [0,x] given the function f(x)=Sigma 10^nx^n from n=[0,oo)?
1 Answer
Explanation:
We have:
f(x) = sum_(n=0)^oo \ 10^nx^n
And so:
f'(x) = d/dx sum_(n=0)^oo \ 10^nx^n
" " = sum_(n=0)^oo \ d/dx10^nx^n
" " = sum_(n=0)^oo \ 10^nd/dxx^n
" " = sum_(n=0)^oo \ 10^n nx^(n-1)
" " = sum_(n=0)^oo \ n10^n x^(n-1)
And:
int_0^x \ f(t) \ dt= int_0^x \ sum_(n=0)^oo \ 10^nt^n \ dt
" "= sum_(n=0)^oo \ int_0^x \ 10^nt^n \ dt
" "= sum_(n=0)^oo \ 10^n \ int_0^x \ t^n \ dt
" "= sum_(n=0)^oo \ 10^n \ [ t^(n+1)/(n+1) ]_0^x
" "= sum_(n=0)^oo \ 10^n \ { x^(n+1)/(n+1) - 0 }
" "= sum_(n=0)^oo \ 10^n \ x^(n+1)/(n+1)
" "= sum_(n=0)^oo \ 10^n / (n+1) \ x^(n+1)
