Find:
limn→∞1n∫∞1dxx2log(1+xn)
The sequence 1nx2log(1+xn)=1x3log(1+xn)xn converges pointwise to 1x3. So if we could apply Lebesgue's Dominated Convergence Theorem, we have:
limn→∞1n∫∞1dxx2log(1+xn)=limn→∞∫∞1dxx3=12
I have a problem with finding a majorant. Could someone give me a hint?
Answer
I showed in THIS ANSWER, using only Bernoulli's Inequality the sequence (1+xn)n is monotonically increasing for x>−n.
Then, we can see that for x≥1 and n≥1, the sequence fn(x) given by
fn(x)=nlog(1+xn)
is also monotonically increasing. Therefore, a suitable dominating function is provided simply by the inequality
1nlog(1+xn)≤1log(1+x)≤1log(2)
Therefore, we have
1nx2log(1+xn)≤1x2log(2)
Using the dominated convergence theorem, we can assert that
limn→∞∫∞11nx2log(1+xn)dx=∫∞1limn→∞(1nx2log(1+xn))dx=∫∞11x3dx=12.
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