# How to evaluate $\lim_{q\to \infty} \left(\sum_{i=1}^m p_i^{q}\right)^{\frac{1}{1-q}}$?

Here is the problem. I have to compute the limit:

$$\lim_{q\to \infty} \left(\sum_{i=1}^m p_i^{q}\right)^{\frac{1}{1-q}}$$

where $p_i$s are numbers from $0$ to $1$ and $\sum_{i=1}^mp_i=1$. I found that solution should be $\frac1{p_{max}}$ but I don't know why.

Thanks a lot for your time.

• We have that $$\Bigl(\sum_{i=1}^m p_i^q\Bigr)^{1/(1-q)}=\frac1{\|p\|_q\cdot\|p\|_q^{1/(q-1)}}$$ and $\|p\|_q\to\|p\|_\infty$ as $q\to\infty$, where $\|\cdot\|_q$ is the $\ell^q$ norm and $\|\cdot\|_\infty$ is the maximum norm. But I'm not sure how to conclude that the limit is actually $1/\|p\|_\infty$. – Cm7F7Bb Nov 11 '15 at 11:01

Use the inequality $$p_{\text{max}}^q\le\sum_{i=1}^m p_i^{q}\le m\,p_{\text{max}}^q.$$