Comparison Test
You can understand
Comparison Test
intuitively as aSandwich Test
.
THIS TEST IS GOOD FOR RATIONAL EXPRESSIONS
.
Direct Comparison Test
Assume that we have a series a_n
, and we're to make up a similar series to it as b_n
:
The logic is:
- If
b > a
&b
converges, thena
converges as well. - If
a > b
&b
diverges, thena
diverges as well.
It’s so much easier if you think it graphically.
▼Refer to video: Comparison Test (KristaKingMath)
▼Refer to xaktly: Comparison Test
Example
Solve:
- Assume the asked series as
a_n
, and we make up a very similar series bigger than it, asb_n
:
- Apply the
p-series test
we get that theb_n
converges. - Since
a_n < b_n
, so according to theSandwich test (Direct comparison test)
,a_n
converges as well.
Limit Comparison Test
Limit comparison test
is like an extension when the Direct comparison test
won't work.
etc., when we compare a
with b
, although b
converges but a > b
, so we can't make any conclusion.
And that's where the limit comparison test
comes in place.
The logic is:
- Take the limit of the division
a/b
. - If the
Limit > 0
, then they both converges or both diverges. - If the
Limit ≤ 0
, then there's no conclusion.
►Jump over to Khan academy for practice: Limit comparison test
▼Refer to video: Limit Comparison Test (KristaKingMath)
▼Refer to xaktly: Limit Comparison Test
Example
Solve:
Example
Solve:
- We can’t easily tell in the comparison who’s greater, so we decide to apply the
limit comparison test
: