x_0=0
x_1=1
x_(n+1)=(x_n+x_(n-1))/2
{Please see this:
http://www.forkosh.dreamhost.com/mimetex...
}
Show that this series is convergent and find its limit.
Additional Details
6 months ago
I should have used %26quot;SEQUENCE%26quot; not series.
Best Answer
Characteristic equation is 2r² - r - 1 = 0, the roots are 1 and -1/2, so the solution is a linear combination of 1^n and (-1/2)^n with 2 constants, taking into account the initial conditions for the nth term of this sequence the expression is:x_n = (2/3)*(1 - (-1/2)^n)
Having |-1/2| %26lt; 1 the limit exists and is equal to 2/3:
lim[n→∞](-1/2)^n = 0, so lim[n→∞] x_n = 2/3
EDIT (after having read Super Man%26#039;s answer): on the text line above I%26#039;ve written
lim[n→∞] x_n = 2/3, but NOTHING about
∑[n=0, ∞] x_n - this series is obviously divergent (so its sum can not be 10/3), the sensible question is about the SEQUENCE {x_n}, the question about the series ∑[n=0, ∞] x_n is NOT INTERESTING. The whole evidence (the way the problem is presented, the link, etc.) suggests the author has meant just that, but there are more intelligent ways to express someone%26#039;s overall opinion on the subject, or to show a sense of humor, or to find out about author%26#039;s original intentions when in doubt, as Jeff Lebowsky has done above.
Other Answers (3)
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the series is 0,1,.5,.75... so it is diverging.
its sum = 10/3 -
answer by pranay c is correct. top contributer Duke is wrong.
the series is : 0, 1, 0.5, 0.75, and so on, no doubt.
if u add this series till 5 terms u get : 0+1+.5+.75+.625 which equals2.875
ans 2/3 = 0.66 is absolutely wrong. which idiots gave him thumbs up? -
It%26#039;s not a series right? You are not summing each term are you? I am assuming this is a sequence, not a series.
Limit of this sequence as n approaches infinity is equal to (1/3) the dude