The limit comparison test says that:

- If the limit of
a[n]/b[n] is positive, then the sum ofa[n] converges if and only if the sum ofb[n] converges.- If the limit of
a[n]/b[n] is zero, and the sum ofb[n] converges, then the sum ofa[n] also converges.- If the limit of
a[n]/b[n] is infinite, and the sum ofb[n] diverges, then the sum ofa[n] also diverges.

For the first one, compare it to . , so the sum converges because does.

For the second one, compare it to . , so the sum diverges because does.