Let 0 < y 1 < x 1 and set a) Prove that 0 < y n
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Let 0 < y1 < x1 and set
a) Prove that 0 < yn < xn for all n e N.
b) Prove that yn is increasing and bounded above, and that xn is decreasing and bounded below.
c) Prove that 0 < xn+1 – yn+1 < (x1 – y1)/2n for n ∊ N.
d) Prove that limn→∞ xn = limn→∞ yn. (This common value is called the arithmetic–geometric mean of x1 and y1.)
Transcribed Image Text:
xn + yn 2and nnEN.
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a This follows immediately from Exercise 126 b By a x n1 x n y n 2 2x n 2 x n Thus y n1 x n1 x 1 ...View the full answer
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