# Inequality involving the k:th root of real sequences $a_k$ and $b_k$ for k large

Let $$a_k \geq 0$$ and $$b_k \geq 0$$ be real sequences and suppose that $$\limsup\limits_{k \rightarrow \infty} \ \sqrt[k]{a_k} \leq 1$$ and $$\limsup\limits_{k \rightarrow \infty} \ \sqrt[k]{b_k} \leq 1$$. Is it true in general that $$\limsup\limits_{k \rightarrow \infty} \sqrt[k]{a_k+b_k} \leq 1$$ ?

From the assumptions, $$a_k<(1+c)^k$$ and $$b_k<(1+c)^k$$ for any $$c > 0$$ for large enough $$k$$.

Therefore $$a_k+b_k<2(1+c)^k$$ so

$$\begin{array}\\ \sqrt[k]{a_k+b_k} &\le 2^{1/k}(1+c)\\ &< (1+c)^2 \qquad\text{for }2^{1/k} < 1+c\\ &< 1+3c \qquad\text{for small enough }c\\ \end{array}$$

This does what you want.

Note that this works for any finite number of summands. What you want, for $$m$$ summands ($$m=2$$ here), is $$m^{1/k} < 1+c$$ or $$k > \dfrac{\ln m}{\ln(1+c)}$$.

• Great Marty - thanks! Oct 31 '18 at 21:22