Find $$\lim _{n \to \infty} \int_0 ^ {1} \left(\frac { 2nx^{n-1}}{1+x}\right) \mathrm{d}x$$
How to find it? The sequence of function is certainly not uniformly convergent. Should I do it in brute force manner?
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1Integrate by parts, ($1/(1+x)$ being the part you differentiate). – Aphelli Jan 25 '19 at 12:23
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After integrating by parts as mentioned above by Mindlack, apply Dominated Convergence Theorem. – Euduardo Jan 25 '19 at 12:39
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This is a typical approximation-to-the-identity problem. Roughly speaking, $nx^{n-1}$ approximates the Dirac delta $\delta(x-1)$, and so, the limit is $\frac{2}{1+x}$ evaluated at $x=1$. – Sangchul Lee Jan 25 '19 at 13:17