Investigating the convergence of a sequence

[Elysian]

Homework Statement

Study the convergence of the following sequences

a_{n} = \int^{1}_{0} \frac{x^{n}}{1+x^{2}}

b_{n} = \int^{B}_{A} sin(nx)f(x) dx

The Attempt at a Solution

For the first one, I said it was convergent. I'm not exactly sure why though, my reasoning was something like this

for all values n>0, the sequence is increasing. It seems to be bounded by 1/2? (not sure here, the integral is messing me up)an increasing sequence that is bounded is also convergent.

The second one, I'm not even sure where to start. Maybe an integration by parts which would end up to repeat itself, as that's what I think I see.

 

[jbunniii]

For a_n, notice that
0 \leq \frac{x^n}{1 + x^2} \leq x^n
for all 0 \leq x \leq 1.

For b_n, are there any constraints on f(x)?

 

[hedipaldi]

for bn:
step1:take f(x) to be any constant an show that the limit is 0.
step2:take f(x) to be any piecewise constant on [A,B] and use step1 to show that the limit is again 0.
step3:take f(x) to be any Riemann integrable function on [A,B],use the definition of the Riemann integral for the existence of a sequence of piecewise constant functions gn(x) such that the integral of f(x)-gn(x) tends to 0,and prove (using the preceding step,the triangle inequality and bound for sin ) that the limit of bn is 0.