Consider the product
.
If you expand it, you get what you are looking for, plus because of the term .
Now can you find an expression for this product?
Write down the numbers 2, 3, . . . , 100, together with
their products taken two at a time, the products taken three at a time, and
so on up to and including the product of all 99 of them. Find the sum of
the reciprocals of the numbers written down.