Sum-the-series-to-10-th-terms-1-1-x-1-1-x-1-1-x-equal-to-

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}} + \dots

e q u a l t o___

Answered by liberty last updated on 18/Nov/20

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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