CPM Homework Help : CALC3RD Problem 9-45

A variant of Ying’s method (manipulating an infinite sum so that the sum appears as a part of itself) can be used in other situations. For example, to evaluate the infinite “nested radical” below, let:

$S = \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \ldots } } } }$

Explain why $S =\sqrt { 2 + S }$ . Then solve for $S$ .

Step 1:

$S^2=2+S$

Step 2:

$S^2-S-2=0$

Step 3:

$(S-2)(S+1)=0$

Step 4:

$S-2=0\text{ or }S+1=0$

Which sum makes sense?