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limits problems......

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limits problems...... Empty limits problems......

Post  mohit Wed Jul 20, 2011 10:59 am

Calculate
limits problems...... Gif.latex?\mathbf{\lim_{n\rightarrow%20\infty}\frac{1+3+6+.........+\frac{n

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limits problems...... Empty Re: limits problems......

Post  Admin Wed Jul 20, 2011 11:31 am

limits problems...... Gif.latex?\hspace{-15}\mathbf{\lim_{n\rightarrow%20\infty}\frac{1+3+6+.........+\frac{n.(n+1)}{2}}{n^3}}\\\\\\%20$\displaystyle\mathbf{\lim_{n\rightarrow%20\infty}\frac{\sum_{r=1}^{n}\frac{r.(r+1)}{2}}{n^3}=\lim_{n\rightarrow%20\infty}\frac{1}{2}.\frac{\sum_{r=1}^{n}\left(r^2+r\right)}{n^3}=\lim_{n\rightarrow%20\infty}\frac{1}{2}.\frac{\sum_{r=1}^{n}.r^2+\sum_{r=1}^{n}.r}{n^3}}$\\\\\\%20$Now%20$\mathbf{\sum_{r=1}^{n}r^2=\frac{1}{6}.n.(n+1).(2n+1)}$\\\\\\%20and%20$\mathbf{\sum_{r=1}^{n}r=\frac{1}{2}.n.(n+1)}$\\\\\\%20put%20values%20in%20Limit,We%20Get\\\\\\%20$\mathbf{=\frac{1}{2}.\lim_{n\rightarrow%20\infty}\frac{\frac{1}{6}.n.(n+1).(2n+1)+\frac{1}{2}.n.(n+1)}{n^3}}$\\\\\\%20$\mathbf{=\frac{1}{2}
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