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Consider a function \( f(\mathrm{n})=\frac{1}{1+\mathrm{n}^{2}} \). Let \( \alpha_{\mathrm{n}}=\frac{1}{\mathrm{n}} \sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{f}\left(\frac{\mathrm{r}}{\mathrm{n}}\right) \) and \( \beta_{\mathrm{n}}=\frac{1}{\mathrm{n}} \sum_{\mathrm{r}=0}^{\mathrm{n}-1} \mathrm{f}\left(\frac{\mathrm{r}}{\mathrm{n}}\right) \) for \( \mathrm{n}=1,2,3, \ldots . \).
\( \mathrm{P} \)
W)
Also \( \alpha=\operatorname{Lim}_{n \rightarrow \infty} \alpha_{n} \& \beta=\operatorname{Lim}_{n \rightarrow \infty} \beta_{n} \). Then prove (a) \( \alpha_{n}\beta_{n} \)
(b) \( \alpha=\beta \)
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