jac006
Golden Master
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- 5,810
well, she is gone, (chick) but she did challenge me. She asked me this question: Let xn and yn be two sequences with the following properties:
>x_(n+1)=xn2+yn2 and y_(n+1)=2*xn*yn for any n>=1 and x1 and y1>0.
>Prove that the series zn=xn/yn converges.
and....
I found that every term zn is greater than 1 so the sum of the series
must diverge.
x(n+1)
z(n+1) = ------- > 1 (aassume this to be true)
y(n+1)
xn2 + yn2
----------- > 1
2.xn.yn
xn2 + yn2 > 2.xn.yn
xn2 - 2.xn.yn + yn2 > 0
(xn - yn)2 > 0
This is always true since a perfect square must be positive. It
follows that z(n+1) > 1.
If the zn terms are all greater than 1 the series must diverge. Yes!!!
>x_(n+1)=xn2+yn2 and y_(n+1)=2*xn*yn for any n>=1 and x1 and y1>0.
>Prove that the series zn=xn/yn converges.
and....
I found that every term zn is greater than 1 so the sum of the series
must diverge.
x(n+1)
z(n+1) = ------- > 1 (aassume this to be true)
y(n+1)
xn2 + yn2
----------- > 1
2.xn.yn
xn2 + yn2 > 2.xn.yn
xn2 - 2.xn.yn + yn2 > 0
(xn - yn)2 > 0
This is always true since a perfect square must be positive. It
follows that z(n+1) > 1.
If the zn terms are all greater than 1 the series must diverge. Yes!!!