9.5 Part 1 Ratio and Root Tests9.5 Part 1 Ratio and Root Tests Absolute ConvergenceAbsolute Convergence
diverges
You’ve all seen the Harmonic Series
01
lim nn
which only proves that a seriesmight converge.
which
1
1
n nconverges
?
What if we were to alternate the signs of each term?
despite the fact that
...6
1
5
1
4
1
3
1
2
1
1
1
How would we write this using Sigma notation?
This is called anThe signs of the terms alternate.
Does this series converge?.
Notice that each addition/subtraction is partially canceled by the next one.
4
1
3
1
6
1
5
1
…smaller and smaller…
The series converges as do all Alternating Series whose terms go to 0
Alternating Series
Alternating Series The signs of the terms alternate.
This series converges by the Alternating Series Test.
This is called the Alternating Harmonic Series which as you can see converges…unlike the Harmonic Series
Alternating Series Test
For the series:
1
11n
nn a
If nn aa 1 0lim n
naand
Then the series converges
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
is what kind of a series?
1
1
3
1
n
n Very good! A geometric series.
What is the sum of this series? 75.04
3
Try adding the first four terms of this series:
27
1
9
1
3
11
3
1
1
1
n
n
740741.27
20
Which is not far from 0.75…but how far?
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
Which means what?
27
1
9
1
3
11
3
1
1
1
n
n
Which is not far from 0.75…but how far?
108
175.740741.
...729
1
243
1
81
1
3
1
108
1
5
1
n
n
Can we agree that ?81
1...
729
1
243
1
81
1
740741.27
20
Alternating Series Estimation TheoremFor 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…
leads us to this:81
1...
729
1
243
1
81
1
San
nn
1
1
If k terms were generated then the error would be no larger than term k + 1
k
k
nn
n Sa 1
1 where k < and
then 1 kk aSS in which is the truncation error
SSk
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: 81
1
27
1
9
1
3
11
3
1
1
1
n
n
Which is bigger than the actual error which we determined to be 108
1
leads us to this:81
1...
729
1
243
1
81
1
Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
That is why it is commonly referred to as the Error Bound
Answer: 81
1 So what does this answer tell us? It simply means that the error will be no larger than
81
1
Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
Since each term of a convergent alternating series moves the partial sum a little closer to the limit:
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!
Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.
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
if the series converges.1L
if the series diverges.1L
if the series may or may not converge.1L
nnaFor the series L
a
a
n
n
n
1limif then
This test is ideal for factorial terms
This also works for the nth root of the nth term. This is called the ROOT TEST(Ex. 57 pg. 508)
if the series converges.1L
if the series diverges.1L
if the series may or may not converge.1L
nnaFor the series Lan
nn
limif then
If the series formed by taking the absolute value of each term converges, then the original series must also converge.
Absolute Convergence
If converges, then we say converges absolutely.na na
If converges, then converges.na na
“If a series converges absolutely, then it converges.”
Conditional Convergence
If diverges, then diverges.nana
If converges but diverges, then we say that
na nana converges conditionally
The Alternating Harmonic Series is a perfect example of Conditional Convergence