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Proving a recursive sequence defined by an integral converges
I want to show that the following sequence of real numbers converges.
I’ve been given the definition:
$$\{a_n\} = \left\{ \int_{0}^{n}\left(1 + \frac{x+x^2+\dots+x^n}{n}\right)dx\right\} $$
I’m not given any tools, it could be a hint.
My attempt:
Using the definition above I could define the limit by
$$\lim_{n\rightarrow \infty}\left(1 + \frac{x+x^2+\dots+x^n}{n}\right) = 1$$
Then by monotone convergence theorem, I would have
$$\lim_{n\rightarrow \infty} \int_{0}^{n}\left(1 + \frac{x+x^2+\dots+x^n}{n}\right)dx = \int_{0}^{\infty}(1+x) dx$$
which is still undefined. Am I on the right track? Or am I missing something obvious?
A:
$$
\frac{1+x+\ldots+x^n}{n}=\int_0^1\left(1+\frac{t}{n}x\right)^ndt
$$
Hence,
$$
\int_0^n\frac{1+x+\ldots+x^n}{n}=\int_0^1\left(1+\frac{t}{n}\right)^n dt
$$
Now, use squeeze theorem.
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