Resolving a list of exercise these two exercises I could not
Determined using the comparison test
1)
2)
The limit comparison test says that:
- If the limit of a[n]/b[n] is positive, then the sum of a[n] converges if and only if the sum of b[n] converges.
- If the limit of a[n]/b[n] is zero, and the sum of b[n] converges, then the sum of a[n] also converges.
- If the limit of a[n]/b[n] is infinite, and the sum of b[n] diverges, then the sum of a[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.
Not necessarily. If you are comparing to and you know that converges, for all but finitely many will tell you that converges.
For example, say converges. You compare to and find:
1.) does not necessarily converge.
2.) will converge.
Note that my choice of 1000 was arbitrary.