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Question

Let $$r=\dfrac{1}{2}$$, consider $$\displaystyle\sum_{n=0}^{\infty}nr^{n}$$ for increasing value of $$n$$ and then find its value as $$n$$ tends to infinity.


A
Tends to 1
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B
Tends to 0
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C
Tends to 2
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D
None of the above
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Solution

The correct option is D Tends to $$2$$
Given:
$$r=\dfrac{1}{2}$$
Let  $$S=\displaystyle\sum_{n=0}^{\infty}nr^{n}$$                        ......(1)
$$\therefore rS=\displaystyle\sum_{n=0}^{\infty}nr^{n+1}$$                      .....(2)
eq (1)- eq (2)
$$(1-r)S=r+r^{2}+r^{3}+....=\dfrac{r}{1-r}$$
$$S=\dfrac{r}{(1-r)^{2}}=2$$
Hence, $$\displaystyle\sum_{n=0}^{\infty}nr^{n}$$ tends to $$2$$.

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