## Examples 4.2.16(d):

Here is an alternative proof of the ratio test that uses the root
test directly. Note that this shows that the root test is 'better' than
the ratio test, because the ratio test can be deduced from the root test.

### Proof:

First, recall the ratio testifand the root testlim sup | athen the series converges absolutely._{n+1}/ a_{n}| < 1

ifWe will use the root test to prove the ratio testlim sup | athen the series converges absolutely._{n}|^{1/n}< 1

Assume that
*lim sup | a _{n+1} / a _{n} | < 1*.
Using the properties of the limit superior, there exists a number

*c*with

*0 < c < 1*, such that

for| a_{n+1}/ a_{n}| < c

*n > N*or equivalently,

for| a_{n+1}| < c | a_{n}|

*n > N*. Then, for any positive integer

*k*, we have that:

for all| a_{N+k}| c | a_{N+k-1}| c ( c | a_{N+k-2}|) ... c^{k}|a_{N}|

*k > 1*. Making the substitution

*N + k = n*, for

*n > N*, this is equivalent to:

or| a_{n}| c^{n-N}| a_{N}|

Taking the

*lim sup*on both sides as

*n*approaches infinity (

*N*is fixed) we obtain, using the result on the

*n*-th Root Sequence:

Hence, the sequence satisfies to root test, and therefore the series converges absolutely.lim sup | a_{n}|^{1/n}c < 1

The proof of divergence is left as an exercise.