Thread: Series => Convergent or Divergent? (I get 2/5 wrong)

1. Series => Convergent or Divergent? (I get 2/5 wrong)

"Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

1.
2.
3.
4.
5. "

The above is what I did. But according to my school's computer system; 1 and 5 are wrong ... how/why?

My logic for #1:
The denominator gets super large so it's like adding 0s in the end so it should converge at some point.

My logic for #2:
The denominator again gets larger than the numerator and should eventually be adding 0s.

What's wrong with my logic?

Any input would be greatly appreciated!

P.S.
1. C (wrong)
2. C
3. D
4. C
5. C (wrong)

2. Do you know that

$\sum_{n=1}^{\infty} \frac{1}{n}$ diverges?

EDIT: Ill elaborate a bit. You can compare that with the second one you got wrong to see why it will diverge.

For the first one use the integral test...

If $\int_2^{\infty} \frac{8}{3n\ln(n)} dn < \infty$ then the series converges, if not then it diverges. (Note the function you integrate must be monotonically decreasing)

$\int_2^{\infty} \frac{8}{3n\ln(n)} dn = \bigg{[}\frac{8}{3}\ln(\ln(n))\bigg{]}_{n=2}^{\infty}$.