The range of `f(x)=[1+sinx]+[2+s in2/x]+[3+s in x/3]++[n+s in x/n]AAx in [0,pi]` , where [.] denotes the greatest integer function, is, `{(n+n-2^2)/2,(n(n+1))/2}` `{(n(n+1))/2}` `{(n^2+n-2^)/2,(n(n+1))/2(n^2+n+2)/2}` `[(n(n+1))/2,(n^2+n+2)/2]`

Explanation

`f(x)=[1+sinx]+[2+"sin"(x)/(2)]+[3+"sin"(x)/(2)]+...+[n+"sin"(x)/(n)] `
for, ` x in (0, pi)`
` sinx, "sin"(x)/(2),"sin"(x)/(3),"sin"(x)/(4),"sin"(x)/(5), … "sin"(x)/(n) in (0,1)`
` :. " for " x in (0,pi)`
`f(x)=1+2+3+ …+ n=(n(n+1))/(2)`
For `x=0, f(x)=1+2+3+...+n=(n(n+1))/(2)`
for ` x=pi, f(x)=1+[2+1}+3+4+...+n`
`=(1+2+3+...+n)+1`
`=(n(n+1))/(2)+1=(n^(2)+n+2)/(2)`
` :. " Range is " {(n(n+1))/(2),(n^(2)+n+2)/(2)}`