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

Question

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 $\limsup\limits_{k \rightarrow \infty} \sqrt[k]{a_k+b_k} \leq 1$ ?

Answer 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)} $.