∑r=0n (−1)r nCr1+rx1+nx is equal to - Infinity Learn by Sri Chaitanya

A
1
B
-1
C
n
D
0

Solution:

Let $\begin{array}{l} E = \sum_{r = 0}^{n} (- 1)^{r n} C_{r} (\frac{1 + rx}{1 + nx}) \\ = (\frac{1}{1 + nx}) \sum_{r = 0}^{n} (- 1)^{r n} C_{r} (1 + rx) \\ = (\frac{1}{1 + nx}) \{\sum_{r = 0}^{n} (- 1)^{r} \cdot^{n} C_{r} + x \sum_{r = 0}^{n} r (- 1)^{r^{n}} C_{r}\} \\ = (\frac{1}{1 + nx}) (0 + 0) = 0 \\ [∵^{n} C_{0} -^{n} C_{1} +^{n} C_{2} -^{n} C_{3} + \dots {(- 1)^{n}}^{n} C_{n} = 0] \end{array}$

$\sum_{r = 0}^{n} (- 1)^{r} (^{n} C_{r}) \frac{1 + rx}{1 + nx}$ is equal to

Solution:

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