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The sequence is defined as

$${a_{n}} = (x^n+y^n)^\frac{1} {n}$$ where $0 \le x \le y$ , and I want to find the limit of this sequence.

I am not sure whether or not I should look at the sequence under different cases, such as when $0 \le x \le y \le 1$, then the limit of the sequence is 1. But then I don't know how to find the limit when $ 1 \le x \le y $.

Or on the other hand, should I just look at the sequence as a whole.

Please give me some ideas on how to do this, thanks for helping.

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1 Answer 1

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Hint: Note that $a_n = \left(x^n+y^n\right)^{1/n} = \left[y^n\left(\dfrac{x^n}{y^n}+1\right)\right]^{1/n} = y\left[\left(\dfrac{x}{y}\right)^n+1\right]^{1/n}$.

Now it should be easy to figure out what happens as $n \to \infty$.

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  • $\begingroup$ I see now, so would the limit be $y$ ? $\endgroup$
    – Lucy
    Commented Nov 1, 2014 at 23:16
  • $\begingroup$ Yes, that is correct. $\endgroup$
    – JimmyK4542
    Commented Nov 1, 2014 at 23:17

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