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Intuition for the convergence of certain types of series
Math 8 2009, Chris Lyons
Let’s start with a few examples:
Example 1.
Look at the series
∞
X
k
=1
1
k
3
+ 4
k

1
.
(1)
Does this converge or diverge?
Discussion.
Obviously this series (1) is not the same thing as
∞
X
k
=1
1
k
3
.
(2)
If it were, this would be easy, because we know that (2) converges! However, (1)
kind of
looks like (2), in
the following imprecise sense: using
≈
to mean “approximately”, if
k
is very very large, we have
k
3
+ 4
k

1
≈
k
3
,
just because
k
3
grows so much faster than 4
k

1 in the expression
k
3
+ 4
k

1. So this means that we ought
to have
1
k
3
+ 4
k

1
≈
1
k
3
.
Using this imprecise reasoning, we would guess that, since (2) converges, then so does (1).
Now here’s the more precise reasoning for why (1) should converge: because all terms of the series are
positive and because
lim
k
→∞
±
1
k
3
+ 4
k

1
²
/
±
1
k
3
²
= lim
k
→∞
k
3
k
3
+ 4
k

1
= 1
(as you should be able to show!), the limit comparison test says that (1) converges because (2) converges.
Example 2.
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This note was uploaded on 07/19/2010 for the course MA 8 taught by Professor Vuletic,m during the Fall '08 term at Caltech.
 Fall '08
 Vuletic,M
 Math, Calculus

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