# Math Help - show that lim(a^n+b^n)^(1/n) = b if 0<a<b

1. ## show that lim(a^n+b^n)^(1/n) = b if 0<a<b

lim(a^n+b^n)^(1/n) = b if 0<a<b

i don't even know what to say. lost.

*edit*

ok got an idea: i know that (a^n+b^n)^(1/n) < (b^n+b^n)^(1/n) = 2^(1/n)*b, which converges to b

i'm thinking of using the squeeze theorem, i just need to find a lower bound that converges to b..

*edit*

found it: (b^n)^(1/n) < (a^n+b^n)^(1/n)

sorry for spamming... i gotta remember the motto: everything you need is in the problem.

2. $(a^n+b^n)^{\frac{1}{n}} = b(\frac{a^n}{b^n} + 1)^{\frac{1}{n}}$

but $\lim_{n\to +\infty} \frac{a^n}{b^n} = 0$ since $b>a$.

$\Rightarrow \lim_{n \to +\infty} b(\frac{a^n}{b^n}+1)^{\frac{1}{n}} = b(1)^{\frac{1}{n}} = b$