Home › Forums › Math questions › Algebra questions

Find the value of:

Posted May 13th, 2009 by alpha
in
$$2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+.....infinity}}}}}$$

Please explain the method to reach the answer.

‹ quadratic Equation Inequality ›

Comments

answer

On May 13th, 2009 Structure says:

$ x_0=2 $

$$x_1=2+\frac{1}{2}$$
$$x_{n+1}=2+\frac{1}{x_n}$$

This sequence is convergent and has a limit
Passing to the limit you have

$$l=2+\frac{1}{l}$$

or

$$l^2-2l-1=0$$

Solution is

$$l=1+\sqrt 2$$

x0=2.0
x1=2.5
x2=2.4
x3=2.4166666666666665
x4=2.413793103448276
x5=2.414285714285714
x6=2.4142011834319526
x7=2.4142156862745097
x8=2.414213197969543
x9=2.41421362489487
x10=2.414213551646055
x11=2.414213564213564
x12=2.4142135620573204
x13=2.4142135624272734
x14=2.4142135623637997
x15=2.41421356237469
x16=2.4142135623728214
x17=2.414213562373142
x18=2.414213562373087
x19=2.4142135623730963
x20=2.414213562373095
x21=2.414213562373095
x22=2.414213562373095
x23=2.414213562373095
x24=2.414213562373095
x25=2.414213562373095
x26=2.414213562373095
x27=2.414213562373095
x28=2.414213562373095
x29=2.414213562373095
x30=2.414213562373095
x31=2.414213562373095
x32=2.414213562373095
x33=2.414213562373095
x34=2.414213562373095
x35=2.414213562373095
x36=2.414213562373095
x37=2.414213562373095
x38=2.414213562373095
x39=2.414213562373095
x40=2.414213562373095
x41=2.414213562373095
x42=2.414213562373095
x43=2.414213562373095
x44=2.414213562373095
x45=2.414213562373095
x46=2.414213562373095
x47=2.414213562373095
x48=2.414213562373095
x49=2.414213562373095

$$1+\sqrt 2=2.414213562373095$$

What is convergent sequence?

On May 14th, 2009 alpha says:

````Maths````
000
(((())))
///

Back to top