Summary Comparison Test | Calculus II Notes

3.3.3 The Comparison Test
This testis similar to the comparison testfor imposer intge rals



Roughly put, we know asum of larger terms is
bigger than a sum of smaller terms

therefore ifwe know the bigger some converges, the smaller as
must well, ifthe small term

diverges, the larger must as well




Theorem:
Let N be a natural number and k <0

if lan1 KC, for all n = N
and non converges then can converges
if AnrKan>0 for all n<-N and Eodn diverges, then an diverges



example: Ein2 2n +
3
+




this could be found using the integral testbutitwould be too mucheffort



when n is
very large n+ 2n+3*t n2


we known, no converges if 421, nin coverges because 42
=




prtcnt3*n'
2
for any nx1, n2+ 2n+3 > n ...




by the comparisontest, an
n2+ 2n +3
=

and Cr =


,
this tells us
intents converges


Its rare for an by for all n, but more common for an=Kbn for all n




example:, n+cos(n)
n3 V3
-




When his large, ncsosal'ntcos(n)=n
n3xbz -n -
z=n3
An n+ cos(n)
=n
=

=


n3 -
13
we know
it converges so we expecton to as well




to verify this with the comparison test:

lan1= Intcos(n))
find K such that n+cos(r) is smaller than
I for alln.
=




n3 -

73 13 -

Vz
①
factor out the dominantterm outof the numerator and dominator


an n cos(n) +cos(n)
3
= +
=




n
13 -

13
1
1 -




Y3n3

② find constant K such that
Itcosm) is smaller than K for all har



1 -
3n3