[tex]\displaystyle\bf\\i^2=-1\\i^3=i^2\times i=-1\times i=-i\\\\i^4=i^2\times i^2= -1\times(-1)=1\\\\i^{2010}=i^{2008+2}=i^{4\times502+2}=i^{4\times502}\times i^2 =\\=\Big(i^4\Big)^{502}\times i^2=1^{502}\times (-1)=1\times (-1)=\boxed{\bf-1}\\\\\\i^{2011}=i^{2010+1}=i^{2010}\times i^1=-1\times i=\boxed{\bf-i}\\\\\boxed{\bf i^{2010}+i^{2011}=-1-i}[/tex]
