<s>[INST] How to find the sum: $1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . . +(50)^{\frac{1}{3}}$ Can some one help me to find the sum of the following expression?
$$1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . .  +(50)^{\frac{1}{3}}$$
 [/INST] You can bound the sum for any upper limit b.
$\frac{3 b^{4/3}}{4}<\sum _{n=1}^b \sqrt[3]{n}< \frac{3}{4} \
\left(-1+\sqrt[3]{1+b}+b \sqrt[3]{1+b}\right)$
Plugging in b = 50 you would get
$ 138.151<\sum _{n=1}^{50} \sqrt[3]{n}<141.097 $
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