#16: RATIO AND ROOT TESTS

a r ⁿ = a r ⁿ

If a _n = r ⁿ then | a _{n + 1} / a _n | = r and = r. The geometric series converges if r < 1 and diverges if r > 1. These observations led to two new convergence tests.

RATIO TEST

r < 1	implies	a n converges absolutely
r > 1	implies	a n diverges
r = 1	implies	try another test

1. Determine if (-1) ⁿ 2 ⁿ / ( n ( 1 + 3 ⁿ ) ) converges.

1. Find a _{n + 1}
2. Find and simplify.
3. Evaluate the limit of the ratio
4. Does the series converge or diverge?

2. Determine if (-1) ⁿ ( ln n ) / ( n! ) converges.

1. Find a _{n + 1}
2. Find and simplify.
3. Evaluate the limit of the ratio
4. Does the series converge or diverge?

3. HOMEWORK #1: Determine if n! / (3n)! converges.

1. Find a _{n + 1}
2. Find and simplify.
3. Evaluate the limit of the ratio
4. Does the series converge or diverge?

ROOT TEST

r < 1	implies	a n converges absolutely
r > 1	implies	a n diverges
r = 1	implies	try another test

4. Determine if 1 / n ⁿ converges.

1. Find
2. Evaluate the limit of the n^th root
3. Does the series converge or diverge?

5. Determine if (-1) ⁿ ( 1- 1 / n ) ⁿ converges.

1. Find
2. Evaluate the limit of the n^th root
3. Does the series converge or diverge?

6. HOMEWORK #2: Determine if e ^{- n} / n converges.

1. Find
2. Evaluate the limit of the n^th root
3. Does the series converge or diverge?

Remark: Since both these test take the absolute value before the limit is taken, they test for absolute convergence. Remember that if a series is not absolutely convergent, it still can be conditionally convergent. For those series that are conditionally convergent, the limit in both the Ratio and Root test will be equal to 1 or the limit will not exist.

HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS	#1, 2; Section 8.6: 2, 8, 14, 20, 22, 36, 38, 46
SUGGESTED PROBLEMS	Section 8.6: 3, 13, 27, 29, 37, 43, 49, 51, 57, 59

Informative web sites:

• http://archives.math.utk.edu/visual.calculus/6/series.11/ Ratio Test by Larry Husch at University of Tennessee
• http://iq.orst.edu/mathsg/SandS/SeriesTests/index.html Series Department of Mathematics, Oregan State University
• http://www-math.cudenver.edu/~rbyrne/online/flash/series/index.htm Slide Show on Convergence Tests by Roxanne Byrne, University of Colorado at Denver
• http://www.cne.gmu.edu/modules/dau/algebra/series/series_frm.html Series Defense Acquisition University Tutorials

READING ASSIGNMENT BEFORE NEXT WORKSHEET: Sections 8.7 and 8.8