1. 9.5 Part 1 Ratio and Root Tests
Absolute Convergence

2. You’ve all seen the Harmonic Series
converges
which ?
despite the fact that
diverges
which only proves that a series
might converge.
What if we were to alternate the signs of each term?
How would we write this using Sigma notation?

3. This is called an
Alternating Series
The signs of the terms alternate.
Does this series converge?.
Notice that each addition/subtraction is
partially canceled by the next one.
…smaller and smaller…
The series converges as do all Alternating Series whose
terms go to 0

4. Alternating Series Test
The signs of the terms alternate.
This series converges by the Alternating Series Test.
Alternating Series Test If
and
For the series:
Then the series converges
This is called the Alternating Harmonic Series which as
you can see converges…unlike the Harmonic Series

5. Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
Very good! A geometric series.
is what kind of a series?
What is the sum of this series?
Try adding the first four terms of this series:
Which is not far from 0.75…but how far?

6. Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
Which is not far from 0.75…but how far?
Which means what?
Can we agree that

7. Alternating Series Estimation Theorem
leads us to this:
Alternating Series Estimation Theorem
For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
Which means that for an alternating series…
If k terms were generated then the error would be no larger than term k + 1
Stopped here
and
where k < 
in which is the truncation error
then

8. Alternating Series Estimation Theorem
leads us to this:
Alternating Series Estimation Theorem
For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
For example, let’s go back to the first four terms of our series:
What is the maximum error between this approximation and the actual sum of infinite terms?
Answer:
Which is bigger than the actual error which we determined to be

9. Alternating Series Estimation Theorem
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
Alternating Series Estimation Theorem
For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
So what does this answer tell us? It simply means that the error will be no larger than
Answer:
That is why it is commonly referred to as the
Error Bound

10. Alternating Series Estimation Theorem
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
Alternating Series Estimation Theorem
For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
This is also a good tool to remember because it is easier than the LaGrange Error Bound…which you’ll find out about soon enough…
Muhahahahahahaaa!

11. Geometric series have a constant ratio between terms
Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge. This is called the RATIO TEST
For the series if
then
This test is ideal for factorial terms
if the series converges.
if the series diverges.
if the series may or may not converge.

12. This also works for the nth root of the nth term
This also works for the nth root of the nth term. This is called the ROOT TEST(Ex. 57 pg. 508)
then
For the series if
if the series converges.
if the series diverges.
if the series may or may not converge.

13. “If a series converges absolutely, then it converges.”
Absolute Convergence
If converges, then we say converges absolutely.
If converges, then converges.
If the series formed by taking the absolute value of each term converges, then the original series must also converge.
“If a series converges absolutely, then it converges.”

14. Conditional Convergence
If converges but diverges, then we say that
converges conditionally
If diverges, then diverges.
The Alternating Harmonic Series is a perfect example of Conditional Convergence