Sum the series to 10^(th) terms (1/(1+(√x))) + (1/(1−x)) +(1/(1−(√x))) +... equal to __


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Question Number 122608 by bramlexs22 last updated on 18/Nov/20

$S u m t h e s e r i e s t o 10^{t h} t e r m s$ $\frac{1}{1 + \sqrt{x}} + \frac{1}{1 - x} + \frac{1}{1 - \sqrt{x}} + . . .$ $e q u a l t o___$
	Answered by liberty last updated on 18/Nov/20
	$T_1 = ((1−(√x))/(1−x)) ; T_2 = (1/(1−x)) ; T_3 = ((1+(√x))/(1−x)) check { ((T_2 −T_1 = ((√x)/(1−x)))),((T_3 −T_2 = ((√x)/(1−x)))) :} . it follows that it is a AP , so S_(10) = 5(((2−2(√x))/(1−x)) + ((9(√x))/(1−x))) S_(10) = 5(((2+7(√x))/(1−x))) = ((10+35(√x))/(1−x))$
	$T_{1} = \frac{1 - \sqrt{x}}{1 - x}; T_{2} = \frac{1}{1 - x}; T_{3} = \frac{1 + \sqrt{x}}{1 - x}$ $check {\begin{cases} T_{2} - T_{1} = \frac{\sqrt{x}}{1 - x} \\ T_{3} - T_{2} = \frac{\sqrt{x}}{1 - x} \end{cases} . it follows$ $that it is a AP, so S_{10} = 5 (\frac{2 - 2 \sqrt{x}}{1 - x} + \frac{9 \sqrt{x}}{1 - x})$ $S_{10} = 5 (\frac{2 + 7 \sqrt{x}}{1 - x}) = \frac{10 + 35 \sqrt{x}}{1 - x}$
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