Question

How do you know if the series `(1/(2n+1))` converges or diverges for (n=1, ∞) ?

Answer 1

`sum_{n=1}^infty 1/{2n+1} = infty`

Explanation:

By comparison, you can say that `2n+1 ~~ n`. They are asymptotically equivalent because

`lim_{n \to \infty} (2n+1)/n = 2`.

So, the series behaves in the same way of

`sum_{n=1}^infty 1/n`,

which is known to be divergent.