Find Limit of Xn as n Approaches Infinity

dannysaf · 2009-09-17T22:14:13+00:00

Consider the sequence [SIZE="3"]x[SIZE="1"]n in which [SIZE="3"]x[SIZE="1"]n = 1/2([SIZE="3"]x[SIZE="1"]n−1 + (3/[SIZE="3"]x[SIZE="1"]n-1) and [SIZE="3"]x[SIZE="1"]1 = a (a not equals 0). Find lim [SIZE="1"]n →∞ [SIZE="3"]x[SIZE="1"]n

Thread starter dannysaf
Start date Sep 17, 2009
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Limit

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SUMMARY

The limit of the sequence defined by x_n = 1/2(x_n-1 + (3/x_n-1)) as n approaches infinity can be determined algebraically. The key observation is that lim x_n equals lim x_n-1 as n tends to infinity. By setting L = lim x_n and substituting into the equation, the limit can be solved to yield L = √3. This conclusion is reached without the need to demonstrate the existence of the limit explicitly.

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Sep 17, 2009

dannysaf

Consider the sequence xn in which xn = 1/2(xn−1 + (3/xn-1) and x1 = a
(a not equals 0). Find lim n →∞ xn

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Sep 17, 2009

VeeEight

If you do not need to show that the limit actually exists, then you can go straight to finding it algebraically. Notice that the lim x_n is the same as lim x_(n-1) (as n tends to infinite). You can use this to find your limit.