AHSME 1952 Problem 43

The diameter of a circle is divided into $n$ equal parts. On each part a semicircle is constructed. As $n$ becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length: $\textbf{(A) }$ equal to the semi-circumference of the original circle $\textbf{(B) }$ equal to the diameter of the original circle $\textbf{(C) }$ greater than the diameter, but less than the semi-circumference of the original circle $\textbf{(D) }$ that is infinite $\textbf{(E) }$ greater than the semi-circumference