# Find limit

• MHB
I wonder if the limit of the following can be converted into integral or some elegant form as N tends to infinity:
$\sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right)$

If we plot or evaluate the value then it does appear that the series converges very fast.

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$\lim_{N\rightarrow\infty}\frac{x}{2^n}\sin^2\left(\frac{x}{2^n}\right)\rightarrow0\times0^2\rightarrow0$ but as the limit is taken over positive $x$ the limit tends to infinity.
$\lim_{N\rightarrow\infty}\frac{x}{2^n}\sin^2\left(\frac{x}{2^n}\right)\rightarrow0\times0^2\rightarrow0$ but as the limit is taken over positive $x$ the limit tends to infinity.