Question

Question: How do you tell if a partial sum of a series is convergent or divergent?

Answer 1

If you have the formula for an \(n\)-th partial sum, you would take the limit as \(n\to\infty\) of the \(n\)-th term.

Answer 2

limit exist and its finite

Answer 3

so for \[\sum_{n=1}^{\infty} n \div \sqrt{n^2 +4}\] with a partial sum to the 10th term, how to tell if it's convergent or divergent would be finding the limit of \[10\div \sqrt{104}\]

Answer 4

a partial sum is a number, and a number neither converges nor diverges, it is just a number

Answer 5

however since
\[\lim_{n\to \infty}\frac{n}{\sqrt{n^2+4}}=1\] the sum does not converge (i.e. it diverges) since the terms to not go to zero