If f.(0$$,π$$)→R is given by $$f(x)=$$∑$$k=1n$$ [$$1+$$$$sin$$⁡$$kx$$],[x] denotes the greatest integer function, then the range of $$f(x)$$ is

Question

If f.(0$$,π$$)→R is given by $$f(x)=$$∑$$k=1n$$ [$$1+$$$$sin$$⁡$$kx$$],[x] denotes the greatest integer function, then the range of $$f(x)$$ is

Infinity Learn · Accepted Answer

$$f(x)=$$∑$$k=1n$$ $$1+$$[$$sin$$⁡$$kx$$]$$=n+$$[$$sin$$⁡x]$$+$$[$$sin$$⁡$$2x$$]$$+$$⋯$$+$$$$sin$$ nx. If $$kx$$≠π$$2$$ for any $$k=1$$, $$2$$ .........n then $$0$$[$$sin$$⁡$$kx$$]$$=0$$,$$k=1$$,$$2$$…n. i.e. $$f(x)=n$$. If $$kx$$$$=$$π$$2for$$ some k then $$x=$$π$$2k$$, hence $$sin$$⁡x,$$sin$$⁡$$2x$$,…$$sin$$⁡(k$$−$$1)x$$ will lie between $$0$$ and $$1$$ so $$sinjx=01$$≤j≤k−$$1$$; $$sin$$⁡$$kx$$$$=1$$ so $$f(x) can be $$n+1$$ or n.

If $f . (0, π) \to R$ is given by $f (x) = \sum_{k = 1}^{n} [1 + \sin k x], [x]$ denotes the greatest integer function, then the range of $f (x)$ is

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If f.(0,π)→R is given by f(x)=∑k=1n [1+sin⁡kx],[x] denotes the greatest integer function, then the range of f(x) is

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If $f . (0, π) \to R$ is given by $f (x) = \sum_{k = 1}^{n} [1 + \sin k x], [x]$ denotes the greatest integer function, then the range of $f (x)$ is