for all n, thenCalculus
Lesson
51
Comparison
of Series
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Next test of convergence: Comparison (two methods).
Direct Comparison: let for all n, then
if 
converges, then 
converges.
if 
diverges, then 
diverges.
This is basically a squeeze theorem!
Example:
To check, note that it looks like 
,
so that's what we'll compare it to.
Since 
converges
and we know 
,
we also know 
converges.
Example: 
We could compare it to 
,
which diverges.
However, since 
, so the test
doesn't tell us anything.
But, we CAN compare the series to 
diverges and 
for 
, so 
diverges.
There's also the Limit Comparison Test: for 
if 
then 
and 
either both converge or
both diverge.
Examples: 
take limit with 
take limit with 
take limit with 
take limit with harmonic
(generalized harmonic)
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