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Math Help - Sequence Limit

  1. #1
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    Sequence Limit

    Show that if (z_n) = [(a^n + b^n)]^(1/n) where 0 < a < b, then lim(z_n) = b.

    I see that b^n will obviously dominate as n goes to infinity, making it (b^n)^(1/n), but how do I show this with algebra or a theorem about sequences (can you use Squeeze Theorem easily?) ?
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  2. #2
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    Quote Originally Posted by jamesHADDY View Post
    Show that if (z_n) = [(a^n + b^n)]^(1/n) where 0 < a < b, then lim(z_n) = b.

    I see that b^n will obviously dominate as n goes to infinity, making it (b^n)^(1/n), but how do I show this with algebra or a theorem about sequences (can you use Squeeze Theorem easily?) ?
    \displaystyle \left( a^n + b^n \right)^{1/n} = \left( b^n \left[ \frac{a^n}{b^n} + 1\right] \right)^{1/n} = b \left( \left( \frac{a}{b} \right)^n + 1 \right)^{1/n} and remember that 0 < \frac{a}{b} < 1.
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  3. #3
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    b^n \leq a^n + b^n \leq b^n + b^n = 2b^n \implies b\leq (a^n+b^n)^{1/n} \leq b\cdot 2^{1/n} --> use squeeze theorem.
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