## 4.2. Convergence Tests

### Root Test

Consider the series . Then:

Compare this test with the Ratio test. Although
this root test is more difficult to apply, it is better than the ratio
test in the following sense: there are series for which the ratio test give
no information, yet the root test will be conclusive. You can also use the
root test to prove the ratio test , but not visa versa.
- if
*lim sup | a*then the series converges absolutely_{n}|^{1/n}< 1 - if
*lim sup | a*then the series diverges_{n}|^{1/n}> 1 - if
*lim sup | a*, this test gives no information_{n}|^{1/n}= 1

It is important to remember that when the root test gives 1 as the
answer for the *lim sup*, then no conclusion at all is possible.

The use of the lim sup rather than the regular limit has the
advantage that we do not have to be concerned with the existence of a
limit. On the other hand, if the regular limit exists, it is the same as
the *lim sup*, so that we are not giving up anything using the
*lim sup*.

**Proof:**

Assume that
*lim sup | a _{n} | ^{1/n} < 1*:
Because of the properties of the limit superior, we know that there exists

*> 0*and

*N > 1*such that

for| a_{n}|^{1/n}< 1 -

*n > N*. Raising both sides to the

*n*-th power we have:

for| a_{n}| < (1 - )^{n}

*n > N*. But the terms on the right hand side form a convergent geometric series. Hence, by the comparison test the series with terms on the left-hand side will converge absolutely.

The proof for the second case if left as an exercise.