What is the value of the sum 1-1 - -2 - 1-2- -3 - - - 1-8 - -9 -

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Properties of Conjugate of a Complex Number

What is the v...

Question

What is the value of the sum $\frac{1}{(\sqrt{1} + \sqrt{2})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + . . . . + \frac{1}{(\sqrt{8} + \sqrt{9})}$ ?

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\frac{1}{\sqrt{9} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{1}}

\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}

(i . e ., ∠ 1 + ∠ 2 + ∠ 3 + \dots ∠ 8 + ∠ 9)

Find the value of:

(1)

(2)

(3)

(4)

(5)

\frac{1}{1! 10!} + \frac{1}{2! 9!} + \frac{1}{3! 8!} + . . . . . + \frac{1}{1! 10!} = \frac{2}{k!} (2^{k - 1} - 1)

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