11.4 The Ratio and Root Tests !! Rita Korsunsky

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.

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

Example 1 Converges or Diverges?

Example 2 Converges or Diverges?

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

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.

Example 1 Converges or Diverges?

Example 2 Converges or Diverges?