http://www.ck12.org Chapter 8. Infinite Series
We see the limitation of the Ratio Test is when lim
∣∣
∣ana+n^1∣∣
∣does not exist (not∞) or is 1.Example 3(Ratio Test inconclusive) limn→∞n+ (^1) n^11 =limn→∞n+n 1 =1, and limn→∞
(n+^11 )^2
n^12 limn→∞
∣∣
∣aan+N^1∣∣
∣=1 for both
∑^1 nand∑n^12. The former (harmonic series) diverges while the latter converges (by, say, thep−test).
Questions (related to the Ratio Test) What if
- limits of
∣∣
∣ana+n^1∣∣
∣exist separately fornodd andneven, i.e limn→∞∣∣
∣a^2 an 2 +n^1∣∣
∣,limn→∞∣∣
∣aa 2 n^2 −n 1∣∣
∣exist but are different?- limn→∞
∣∣
∣aa^22 nn+− 11∣∣
∣,limn→∞∣∣
∣a^2 an 2 +n^2∣∣
∣exist but are different?Exercises
Determine whether the following series is absolutely convergent, conditionally convergent, or divergent with the
Ratio Test and other tests if necessary:
1.∑∞n= 1 n 23 n
2.∑∞n= 1 e−^2 nn!
3.∑∞n= 1 (− 1 )n−^1 √^1 n
- 3 − 23. 4 + 34. 5 − 45. 6 +...
nth - root Test
If the general termanresembles an exponential expression, the following test is handy.
Theorem(The Root Test)
(A) If limn→∞|√nan|=α<1, then the series is absolutely convergent.
(B) If limn→∞|√nan|=α>1 or limn→∞|√nan|=∞, then the series is absolutely divergent.
(C) If limn→∞
∣∣√n
n∣∣
=α=1, then the test is inconclusive.
The proof is similar to that of the Ratio Test and is left as an exercise.
Example 1Consider∑an=∑n^1 pwherep>0. We already know it is convergent whenp>1. To apply the Root Test,
we need limn→∞√nnwhich is 1 after some work. Alternatively, we could check limn→∞
∣∣
∣ana+n^1∣∣
∣=1 so limn→∞√nn=^1
by the argument similar to the proof of the Ratio Test. The Root Test is also inconclusive.
Example 2Test the series∑( 2 nn++^13 )nfor convergence.
Solution. Letan=( 2 nn++^13 )n. Then limn→∞n
√
|an|=limn→∞ 2 nn++^13 =^12. So the series is absolutely convergent.
What if we apply the Ratio Test?
limn→∞∣∣
∣ana+n^1∣∣
∣=limn→∞(n+ 2
2 n+ 5)n+ (^1) /(n+ 1
2 n+ 3
)n=lim
n→∞
(n+ 2
2 n+ 5
).[(n+ 2 )( 2 n+ 3 )
(n+ 1 )( 2 n+ 5 )
]n
We could still argue the limit is^12 with some work. So we should learn to apply the right test.
Exercises
Determine whether the following series is absolutely convergent, conditionally convergent, or divergent with the
Root Test and other tests if necessary: