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I want to caclulate $$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }$$ for a,b ≥ 0

I wanted to isolate $\sqrt[n]{a^n}$ but I really don't know how to show that the limit is 1.

I know that $\sqrt[n]a \rightarrow 1$ ,so I thought that I seperate the function to $\sqrt[n]{a^n}$ and $\sqrt[n]{b^n}$ and show that they are equally going to the same limit. So that $\sqrt{a^n}$ and $\sqrt{b^n} = \lim{\sqrt[n]{ a^{ n } + b^{ n } } }$.

Jean Marie
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Marie MI
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    Welcome to math stack exchange. The result depends on $a$ and $b$. In general, the limit is $\max(a,b)$. – Peter Dec 07 '19 at 14:39
  • I hope the edit did not change the intent of the question. Please check it. – Peter Dec 07 '19 at 14:55
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    Welcome to MSE. You have to enclose MathJax commands in $ signs before they will be effective. Also, you have to write \ before functions like sqrt. $\sqrt[n]{a^n + b^n}$ is typeset as $\sqrt[n]{a^n + b^n}$ Here is a tutorial – saulspatz Dec 07 '19 at 14:56
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    Several very similar questions, although unless I'm overlooking something, your limit is much simpler --- factor $a^n + b^n$ as $a^n(1 + (b/a)^n)$ or as $b^n((a/b)^n + 1),$ depending on whether $a > b$ or $b > a.$ – Dave L. Renfro Dec 07 '19 at 15:17

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The limit is $\max(a,b)$. Assume WLOG that $a> b$. Then

$$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }=\lim_{n\rightarrow \infty}a{\sqrt[n]{ 1 + \left(\frac{b}{a}\right)^n } }=a$$

since

$$\lim_{n\rightarrow \infty}{\sqrt[n]{ 1 + \left(\frac{b}{a}\right)^n } }=1$$

On the other hand, if $a=b$ then

$$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }=\lim_{n\rightarrow \infty} {\sqrt[n]{2a^n} }=\lim_{n\rightarrow \infty}2^{1/n}a=a$$

Axion004
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  • How would a beginner prove $\lim \sqrt[n]{1 +r^n} =1$ if $0 < r < 1$? (Or for that matter that $\lim\sqrt[n]{1 + r^n} = r$ if $r > 1$). You can't say as $n\to\infty$ that $1+r^n\to 0$ because you can't separate the limits. What's the beginner to do? – fleablood Dec 07 '19 at 19:21
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    @fleablood They could observe that $1 < 1+r^n <2$. – Gunnar Sveinsson Dec 07 '19 at 19:45
  • What did you do in the first line ? I do not understand how sqrt a+b is a * sqrt 1+(b/a)^n. Where is then "n" from a^n? – Marie MI Dec 08 '19 at 20:36
  • ${\sqrt[n]{ a^ n + b^ n } }=(a^n+b^n)^{1/n}=\left(a^n\left(1+\frac{b^n}{a^n}\right)\right)^{1/n}=\left(a^n\left(1+\left(\frac{b}{a}\right)^n\right)\right)^{1/n}$ – Axion004 Dec 08 '19 at 20:43