1. 11.4 The Ratio and Root Tests!!
Rita Korsunsky

2. Ratio Test Let be a positive-term series, and suppose
1) IF L<1, Converges
2) IF L>1, or
Diverges
3) IF L=1,
Apply a different test; the series may be convergent or divergent.

3. So, the series converges if L<1, and diverges if L>1
Let’s try to prove it:
For geom. series,
1) IF L<1, Converges
2) IF L>1, or
and if < 1
Diverges
Apply a different
test; the series may be
convergent or divergent.
3) IF L=1,
approaching to
In part 1)
<1
So, the series converges if L<1, and diverges if L>1
But why is part 3) inconclusive?!
Use Ratio Test for
p-series, p=2>1
convergent
Use Ratio Test for
harmonic
divergent

4. Example 1
Converges or Diverges?

5. Example 2
Converges or Diverges?

6. Example 3 Converges or Diverges? Use a different test
Use Limit Comparison Test with
P-series,p=2>1,
convergent

7. Root Test Let be a positive-term series, and suppose
IF L<1, Converges
IF L>1, or
Diverges
IF L=1,
Apply a different test; the series may be convergent or divergent.

9. Example 1
Converges or Diverges?

10. Example 2
Converges or Diverges?