A Difficult Problem About Series.?

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The series is defined as :
x_0=0
x_1=1
x_(n+1)=(x_n+x_(n-1))/2
{Please see this:
http://www.forkosh.dreamhost.com/mimetex.cgi?x_0=%200,\,x_1=%201,\,x_{n%20+%201}=%20\frac{{x_n+%20x_{n%20-%201}%20}}{2}
}
Show that this series is convergent and find its limit.

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"A Difficult Problem About Series.?" was posted on Friday, July 10th, 2009 at 6:37 am.

4 Responses to “A Difficult Problem About Series.?”

~AN INDIAN~(2009-07-10):

Wrong category
LIZ(2009-07-10):

The series is convergent and the limit is one. Simply use the method of induction.
Garrett H(2009-07-10):

Closed form expression for x(n):
x(n) = 2/3 – 2/3 (-1/2)^n
Proof:
x(0) = 2/3 – 2/3 = 0 – correct
x(1) = 2/3 + 1/3 = 1 – correct
Induction:
x(n+1) =
= 1/2 [2/3 - 2/3 (-1/2)^n + 2/3 - 2/3 (-1/2)^(n-1)] =
= 2/3 – 1/3 [(-1/2)^n + (-1/2)^(n-1)] =
= 2/3 – 1/3 (-1/2)^(n+1) [4 - 2] =
= 2/3 – 2/3 (-1/2)^(n+1)
Answer: the limit is 2/3
feeju(2009-07-10):

isnt this mathematics