11.6
• Ratio Test:
Let’s assume we’re starting with a positive series,
∞
X
an ,
an > 0,
n=1
and let’s assume we did the ratio test. So we calculated
an+1 = lim an+1
L = lim n→∞
an n→∞ an
and the ratio test was successful and told us that the series was convergent, so that
means
L < 1.
Now we can’t get too deep into the real definition of a limit, but the limit is just
an+1
gets really close to L.
telling us that when n gets really, really big, our fraction
an
How close to L? As close as I like. For example, let’s pick any real number r between
L and 1,
L<r<1
I can definitely get that fraction closer to L than r is. What I mean is, if I choose n
big enough, I can be sure that
an+1
< r < 1.
an
In math speak, I say that there is a positive integer K, so that for all the n ≥ K,
an+1
< r < 1.
an
Now let’s look at the tail of our series, starting at this number I chose, K:
∞
X
an
n=K
1. If the tail,
∞
X
an
n=K
converges, then
∞
X
an
n=1
converges. Why? (We don’t need anything formal, just try to explain it.)
2. Write out the first four or five terms of the series
∞
X
n=K
an =
3. Now use the fact that for n ≥ K
an+1
<r<1
an
to find a relationship between the first two terms of this series.
Instructor: The students should arrive at
aK+1
<r
aK
⇒
aK+1 < raK
4. Now use that inequality to find a similar relationship between the second and
third terms, aK+1 and aK+2 .
5. Now combine what you learned in part 3 and part 4 to get a relationship
between the first and third terms, aK and aK+2 .
6. Similarly, use a relationship between the third and fourth terms to get a
relationship between the first and fourth terms.
7. Can you guess what the relationship is between the first and fifth terms, aK and
aK+4 ?
8. Use this information to get an inequality on your series in part 2,
∞
X
an = aK + aK+1 + aK+2 + aK+3 + · · · ≤
n=K
9. Can you rewrite this inequality in sigma notation?
∞
X
an ≤
n=K
10. Factor an aK out of the series on the right. What kind of series are you left with?
11. Is this remaining series convergent or divergent? Why?
12. What test would you use now to understand our original series (rather, the tail
∞
X
to our original series,
an )?
n=K
13. Is the tail convergent or divergent?
14. What do we now know about the series
∞
X
n=1
an ?
• Root Test:
Let’s assume we’re starting with a positive series,
∞
X
an ,
an > 0,
n=1
and let’s assume we did the root test. So we calculated
p
1
L = lim n |an | = lim (an ) n
n→∞
n→∞
and the root test was successful and told us that the series was convergent, so that
means
L < 1.
Now we can’t get too deep into the real definition of a limit, but the limit is just
1
telling us that when n gets really, really big, the root (an ) n gets really close to L.
How close to L? As close as I like. For example, let’s pick any real number r between
L and 1,
L<r<1
I can definitely get that root closer to L than r is. What I mean is, if I choose n big
enough, I can be sure that
1
(an ) n < r < 1.
In math speak, I say that there is a positive integer K, so that for all the n ≥ K,
1
(an ) n < r < 1.
Now let’s look at the tail of our series, starting at this number I chose, K:
∞
X
an
n=K
1. If the tail,
∞
X
an
n=K
converges, then
∞
X
an
n=1
converges. Why? (We don’t need anything formal, just try to explain it.)
2. Write out the first four or five terms of the series
∞
X
n=K
an =
3. Now use the fact that for n ≥ K
1
(an ) n < r < 1
to find a relationship between the first two terms of this series.
Instructor: The students should arrive at
1
(aK ) K < r
⇒
aK < r K
4. Now use that inequality to find a similar relationship bound on the second term,
aK+1 .
5. Can you get a bound on aK+2 ? And aK+3 ?
6. Use this information to get an inequality on your series in part 2,
∞
X
an = aK + aK+1 + aK+2 + aK+3 + · · · ≤
n=K
7. Can you rewrite this inequality in sigma notation?
∞
X
an ≤
n=K
8. Factor out an rK . What kind of series do you have on the right?
9. Is this remaining series convergent or divergent? Why?
10. What test would you use now to understand our original series (rather, the tail
∞
X
an )?
to our original series,
n=K
11. Is the tail convergent or divergent?
12. What do we now know about the series
∞
X
n=1
an ?