Lebesgue measure theorems – Mathematics Stack Exchange

We are looking at the Lebesgue measure on $mathbb{R}$ and ${(f_n)_{nin mathbb {N}}colon mathbb {R} to overline {mathbb {R}}}$ with $f_n(x) = 1+sum _{k=1}^n |x|^k$.

a. Does $(f_n)_{nin mathbb {N}}$
fulfill the requirements of the dominated convergence theorem?

b. Does $(f_n)_{nin mathbb {N}}$
fulfill the requirements of Fatou’s theorem?

c. Does $(f_n)_{nin mathbb {N}}$
fulfill the requirements of the monotone convergence theorem?

I think that b is true and that c is wrong, is that correct? I’m not really sure about a.