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Q. The value of $\underset{x \rightarrow 1}{lim} \displaystyle \sum _{r = 1}^{10} \frac{x^{r} - 1^{r}}{2 \left(x - 1\right)}$ is equal to

NTA AbhyasNTA Abhyas 2020Limits and Derivatives

Solution:

We have, $\underset{x \rightarrow 1}{lim} \displaystyle \sum _{r = 1}^{10} \frac{x^{r} - 1^{r}}{2 \left(x - 1\right)}$
$=\frac{1}{2} \displaystyle \sum _{r = 1}^{10}r\cdot 1^{r - 1}=\frac{1 + 2 + 3 + \ldots + 10}{2}=\frac{10 \left(10 + 1\right)}{4}=\frac{55}{2}$