Do you even telescope?

Algebra Level 2

Find the value of the following sum:

\[\sqrt{\frac{1}{1^3}}+\sqrt{\frac{1}{1^3+2^3}}+\sqrt{\frac{1}{1^3+2^3+3^3}}+\ldots \infty\]

Note that \(\infty\) here refers to the series having infinite terms and not the value \(\infty\) in the sum! In other words, the sum continues indefinitely.

Inspired by this problem.

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