# How do you show that the series #1+sqrt2+root3(3)+...+rootn(n)+...# diverges?

The series is divergent:

We can also proceed with direct comparison: as the general term of the series is positive we can take its logarithm:

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To show that the series (1 + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} + \ldots) diverges, we can use the comparison test.

We know that (\sqrt{n} > 1) for all (n > 1). Therefore, each term in the series is greater than 1.

Since the series is made up of terms greater than 1, it diverges by the nth term divergence test.

Hence, the series (1 + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} + \ldots) diverges.

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