1. Direct Comparison Tests
Objective: Be able to use the direct comparison test to describe the convergence or divergence of series.
ES: C4 Explicitly assess information and draw conclusions
Warm-Up: Determine the convergence of this series. Include whether it converges conditionally or absolutely if necessary.

2. Direct Comparison Test (DCT)

3. Determine the convergence or divergence of
is a geometric series where r=1/3<1, thus the series converges.
Since we were able to show our original series is less than this series, by the DCT it must also converge.

4. Determine the convergence or divergence of
is a p-series where p=1/2<1, thus the series diverges.
Since we were able to show our original series is less than this series, by the DCT it could still converge or diverge at a slower rate! I can’t tell using this comparison…
bummer...Guess I’ll have to try a different comparison.

5. Determine the convergence or divergence of
Starting with the 5th term, 1/n is smaller than the original forever after. Thus
is a p-series where p=1>1, thus the series diverges.
Since the original series is greater than this series for all infinite terms beyond the 4th term, by the DCT it must diverge.

6. Determine the Convergence or Divergence of each
Determine the Convergence or Divergence of each. Make sure you state what your comparison is and show whether it converges or diverges.