Let F<sub>0</sub>(x)=x, and for each positive integer k,
F<sub>k</sub>(x)=k[∫<sub>0</sub><sup>x</sup>(F<sub>k-1</sub>(t)dt+x∫<sub>0</sub><sup>-1</sup>(F<sub>k-1</sub>(t)dt)]
Prove that for each positive integer k,
(a) F<sub>k</sub>(-1)=F<sub>k</sub>(0)=0
(b) Use induction to prove that
(i) F<sub>k</sub>(x) is a polynomial in x of degree k+1 whose constant term is 0 and the coefficients of x<sup>k+1</sup> and x<sup>k</sup> are respectively 1/(k+1) and 1/2
(ii) F<sub>k</sub>(x)-F<sub>k</sub>(x-1)=x<sup>k</sup>
(c) If n is any positive integer, show that
F<sub>k</sub>(n)=1<sup>k</sup>+2<sup>k</sup>+3<sup>k</sup>+.......+n<sup>k</sup>
<sup>
</sup><sup></sup><sup></sup>
F<sub>k</sub>(x)=k[∫<sub>0</sub><sup>x</sup>(F<sub>k-1</sub>(t)dt+x∫<sub>0</sub><sup>-1</sup>(F<sub>k-1</sub>(t)dt)]
Prove that for each positive integer k,
(a) F<sub>k</sub>(-1)=F<sub>k</sub>(0)=0
(b) Use induction to prove that
(i) F<sub>k</sub>(x) is a polynomial in x of degree k+1 whose constant term is 0 and the coefficients of x<sup>k+1</sup> and x<sup>k</sup> are respectively 1/(k+1) and 1/2
(ii) F<sub>k</sub>(x)-F<sub>k</sub>(x-1)=x<sup>k</sup>
(c) If n is any positive integer, show that
F<sub>k</sub>(n)=1<sup>k</sup>+2<sup>k</sup>+3<sup>k</sup>+.......+n<sup>k</sup>
<sup>
</sup><sup></sup><sup></sup>
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