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.
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.
Consider the product
$\displaystyle \left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\righ t)\dots\left(1+\frac{1}{100}\right)$.
If you expand it, you get what you are looking for, plus $\displaystyle 1$ because of the term $\displaystyle 1 \times 1 \times \dots \times 1$.
Now can you find an expression for this product?