Find The Limiting Ratio

Calculus Level 5

Beginning with a square of area 1, construct a sequence of rectangles by alternately adjoining a rectangle of area one alongside or on top of the previous rectangle. First five rectangles are shown:

The limiting ratio of length to height has a closed form. Find this closed form.

Give your answer to 3 decimal places.

Source: Gerald Myerson (Elementary Problems, American Mathematical Monthly).

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