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  4. The value of undersetx arrow 2lim displaystyle ∑ r = 17 (xr - 2r/2 r (x - 2)) is equal to

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Q. The value of $\underset{x \rightarrow 2}{lim} \displaystyle \sum _{r = 1}^{7} \frac{x^{r} - 2^{r}}{2 r \left(x - 2\right)}$ is equal to

NTA AbhyasNTA Abhyas 2020Limits and Derivatives

Solution:

We have, $\displaystyle \sum _{r = 1}^{7}\underset{x \rightarrow 2}{lim} \frac{\left(x^{r} - 2^{r}\right)}{2 r \left(x - 2\right)}$
$=\displaystyle \sum _{r = 1}^{7}\frac{r \cdot 2^{r - 1}}{2 r}=\frac{1 + 2^{1} + 2^{2} + . \ldots .. + 2^{6}}{2}=\frac{127}{2}$