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The Root of the Problem

Age 16 to 18 Short
Challenge Level

Find the sum of $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

Can you invent any similar sums which have integer answers?
Did you know ... ?

Whilst this series can be summed using elementary methods mathematicians devise various ways in which the sums of series can be analysed. These are explored in greater detail in university analysis courses.