Monday 13 May 2013

limit of sequence with 5th root



I need to solve this:
$$
\lim_{n\to\infty} \sqrt[5]{n^5+2n^4}
-\sqrt[5]{n^5-n^4}

$$
I am beginner in calculating limits of sequences. I would be happy if someone could show how to solve it or give me a hint so I could try to work it out by myself. Probably it's not complicated but I don't know how to get rid of this 5th roots.


Answer



Hint:
$$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$$
Let $a=\sqrt[5]{n^5+2n^4}$ and $b=\sqrt[5]{n^5-n^4}$.


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