- #1
cornstarch
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Homework Statement
if lim f(n) = s , prove that lim (f(n))^{1/3} = s^{1/3}. How do you know that as n approaches infinity, f(n) and s have the same sign. n is just an index in this case and f is not a function but a sequence.
The attempt at a solution
so we know that |f(n) - s| < epsilon, using a^3 - b^3 I can factor:
|f(n)^{1/3} - s^{1/3}|*|f(n)^{2/3} + (f(n)*s)^{1/3} + s^{1/3}| < epsilon. Divide both sides and we get |f(n)^{1/3} - s^{1/3}| < epsilon/(|f(n)^{2/3} + (f(n)*s)^{1/3} + s^{1/3}|) which proves the first part. I don't know how to argue that as n approaches infinity, f(n) and s have the same sign.
if lim f(n) = s , prove that lim (f(n))^{1/3} = s^{1/3}. How do you know that as n approaches infinity, f(n) and s have the same sign. n is just an index in this case and f is not a function but a sequence.
The attempt at a solution
so we know that |f(n) - s| < epsilon, using a^3 - b^3 I can factor:
|f(n)^{1/3} - s^{1/3}|*|f(n)^{2/3} + (f(n)*s)^{1/3} + s^{1/3}| < epsilon. Divide both sides and we get |f(n)^{1/3} - s^{1/3}| < epsilon/(|f(n)^{2/3} + (f(n)*s)^{1/3} + s^{1/3}|) which proves the first part. I don't know how to argue that as n approaches infinity, f(n) and s have the same sign.