Weekly Challenge 18: the Root of the Problem
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.