ABSTRACT
Although I do not know of any brute force method for solving this problem, the solution I do know requires the following clever insight. Let x equal the fraction above:
x = + +
+ +
+…
1 1
1 1
1 1
1 1 1
.
Because the fraction has the same repeating form and is infinite in extent, we can replace the entire denominator in the second term by x, and write
x x
= +1 1 .
