I'm having some difficulty with the following exercise:

I'm being asked to find the domain of convergence, $D$, for the following sequence of functions, and find the limit function for each $x \in D$:

$$f_n(x) = \left(\frac{2n^3 + n^2x+2}{2n^3 - n^2 + x^2} \right)^n.$$

I think I should try to turn it somehow into the form of $(1 + \frac{x}{n})^n$ so I can get something similar to $e^x$ after taking the limit, but I haven't managed to do that.

I have tried to look for similar examples on the web, but haven't managed to find any (which may be because I don't know how to look for it good enough).

Help will be much appreciated. Thanks!


Notice that $f_n(x)=(1+\frac{a_n(x)}n)^n$ where $$a_n(x)=\frac{x+1+\frac{2-x^2}{n^2}}{2-\frac1n+\frac{x^2}{n^3}}.$$ Does that help?

  • $\begingroup$ I got it. Thanks! $\endgroup$ – tfreifeld May 11 '17 at 16:12

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