Exam 3 will cover chapter 11. This sheet has three sections. The first section will remind you about techniques and formulas that you should know.  The second gives a number of practice questions for you to work on.  The third section give the answers of the questions in section 2.

Review

Sequences

Sequences are an important part of Chapter 11, because so much of what we do involves them. You need to be able to take a sequence and determine its behavior. Is is increasing or decreasing? Does it converge in the limit? What does it converge to? Some theorems may be of help here:

1. If a sequence converges, it is bounded.

2. If a sequence is bounded and is (eventually) increasing or decreasing, then it converges.

3. If a sequence {an } matches a function f (i.e. f(n) = an ) and

lim  f(x) = L,

n →∞

then the limit of the sequence is also L.

Rule 3 is useful because we can use everything we know about limits of functions to find limits of sequences. Since L’Hopital’s rule is one of them, you should expect to use it.  There are other rules about sums of sequences and products of sequences, etc. You are advised to review them in the text.

Sometimes we can determine whether a sequence is (eventually) increasing or decreasing by looking at a function f(x) with f(n) equalling the nth term of the sequence. If the function is increasing, the sequence is also.

Important Sequences

Some limits occur often enough that it is advisable to know about them in advance.  For example, Dr.  McKay expects all of his students to know the following:

1. If c is a real, positive number, then c1/n  → 1.

2. If c is a real, positive number, then → 0. cn

n!

4. n1/n → 1.

5.  (1 + )n → ec .

All of the above limits except the third can be proven using L’Hopital’s rule.  The third is a bit tricky but can be

c

n + 1

If you encounter these limits in a problem, you are welcome to use what you know about them and move on. (Unless the question *is* one of these limits.)

Recursive Sequences

Most sequences that we deal with have a rule we can apply to find the nth term of the sequence. Recursive sequences, however, only have a rule that allows us to find the nth term if we know all of the other terms that come before it. The most famous recursive sequence is the Fibbonaci sequence given by

a1  = 1,  a2  = 1,  an  = an − 1 + an −2 .

Generally, it is difficult to tell what a recursive sequence does. The theorems listed above can show that a recursive sequence converges. For example if the sequence can be shown to be bounded and is increasing, then it must have a limit. If a recursive sequence converges, then finding the limit of the sequence is easy: Let L represent the limit.

Since the sequence converges to L, every sequence element in the recursion formula converges to L also. For example, suppose we wish to find the limit of

a1  = 1,a2  = 1,an  = an − 1 − .

If we are reasonably sure the limit exists, we can replace an  by L to get

L = L − 0

You can then use this to solve for L.  Make sure the series converges however, or you may arrive at the wrong conclusion. For example, suppose we have the recursive sequence

a1  = 1,an  = 2an − 1 − 1.

Using this technique gives L = 2L − 1, which yields L = 1. However, it is not hard to see that this sequence diverges.

Series

In this section we learned about convergent and divergent series. A series converges if the sequence of partial sums converge. There are some particular types of series that we learned about:

Geometric Series 工 arn . We learned that the geometric series converges to if |r| < 1 and diverges other-

Harmonic Series . We saw by examination of the partial sums s2n   that this diverges. The integral test also

shows the divergence of this series.

Alternating Harmonic Series ( − 1n+1 We know by a later test that the alternating Harmonic series con-

verges. We know by the Maclaurin series of ln(1 + x) that it converges to ln(2).

Telescoping Series This is a series where the partial sum collapses to the sum of a few terms. We can then take the limit of the partial sum to see what the series converges to.

Note that in 11.2 the only series whose sums we could calculate were geometric and telescoping.

Tests for Convergence

We learned about the following tests for convergence:

Divergence Test If an  →\ 0 then 工 an  diverges. This is an excellent test to start with because the limit is often easy to calculate.  Keep in mind, however, if the limit is 0, then the Divergence test tells you nothing.  You must try some other test.

p series If you recognize a series as a p series,

工

then you can use the fact that a p series converges when (and only when) p > 1.

Geometric series We discussed this in the last subsection.

Comparison Test To use the comparison test, we need to have a large group of test series available. We also need to know if these test series converge or not.  The most common test series for the comparison test are the p series and the geometric series.  If the series ”acts like”a p series, or ”acts like”a geometric series, then you may wish to use the comparison test. Remember, if 0 ≤ an  ≤ bn  and

❼ 工 bn  converges, then 工 an  converges.

❼ 工 an  diverges, then 工 bn  diverges.

Limit Comparison Test This test works well for the type of problems that also work with the comparison test, but is somewhat easier.  You still need the test series, but you don’t need to work to make the terms of the series greater than or less than some known series. You only need to check the limit

an

n →∞ bn .

If it is finite and positive, then both series converge or both diverge.  Since you already know about one of them, you then know about the other.

Integral Test If we are trying to determine whether 对 an  converges, and there is a function f(x) with f(n) = an , then the sum converges iff

\a ∞ f(x) dx  converges.

(We assume that both the series {an } and f(x) are positive.)  So the integral test is handy if the associated function can be integrated without too much difficulty.

Alternating Series Test To use the alternating series test, you need to verify three things: The series is alternating. (This can usually be done by inspection). The terms of the series converge to 0. (Hopefully you did this when you applied the Divergence test.) Finally, the terms of the absolute values are decreasing. The second statement does not necessarily imply the third. If this is true, then the alternating series test tells us the series converges.

Ratio Test If

nl ' ' = L,

then the series is absolutely convergent if L < 1 and divergent if L > 1. If L = 1, the test fails. This test works really well when a factorial is present in an .

Root Test If

lim  ^n |an | = L,

then the series is absolutely convergent if L < 1 and divergent if L > 1. If L = 1, the test fails. This test works really well when there are powers of n in an .

Remember, the Integral test and the comparison tests only work when the series has non-negative terms. If you have a series where the terms are both positive and negative, then you must be able to say whether the series converges absolutely, converges conditionally, or diverges. It is one of these. These are mutually exclusive conditions.

Estimating the tail

In an infinite series, the tail is a term usually used to indicate the “last”part of the series. For example, if we wish to approximate the sum of the following convergent series,

,

then we can write it as

+ n

The part that is still an infinite sum is called the tail. The sum of the tail is called the error of our approximation. If we can test convergence of a series by the integral test, then there is an easy way to find an estimate of the tail: Assume f(x) is defined on [b, ∞) for some b, and f(n) = an . Then

\k 1 f(x)dx ≤ n ≤ \k ∞ f(x)dx.

For example suppose that we sum the first 5 terms of the above series:

∞ 4

≈

1 1 1 1

= 1 +      +       +       +        = 1.049324231

How close is this? We find that

\5 ∞ dx = = 0.002066115702,

and

\4 ∞ dx = = 0.003086419753.

Thus, the error is between these two numbers.

If we can use the comparison test to find convergence, then we can sometimes still use the above formula, but only for upper bounds. For example, if I am trying to estimate

之(∞)   n3(n)11 ,

the fact that

<

means that

n n3(n)11 < \k ∞ dx.

Power Series

Recall that a power series is a series of the form

∞

之 an (x − c)n .

The value c is called the center of the power series, and the values an  are called the coefficients.

A power series is a way to represent a function. However, the power series may have a different domain than the function does. To find the domain of the power series, (called the interval of convergence), we do the following:

1. Apply the ratio or root test to the power series.  If the limit is 0, the power series converges everywhere and

the radius of convergence is ∞ . If the limit is ∞ , the power series converges only at the center, and the radius

of convergence is 0. Otherwise, set the limit to be less than 1, and rework the inequality so it says |x − c| < R.

2.  The power series is now guaranteed to converge absolutely on  (c − R,c + R), and diverge on  (−∞ ,c − R) ∪

(c + R, ∞). We now test the power series at the endpoints. Plug the endpoints c − R and c + R into the power

interval of convergence using parentheses to indicate the power series does not converge at an endpoint, and a

bracket to indicate it does.

Finding sums of series

Finding a power series that represents a specific function is the next topic. The first one we learned was the geometric

series:

= 之 xn   iff x ∈ ( − 1, 1).

We then found the sum of several series by differentiating, integrating, multiplying by x, etc.

The Taylor series of a function is

(x − c)n

and can also be used to find the power series of a function.

Notice that the interval of convergence of these series is still very important. We need to know when we can trust them.

In addition to the geometric series above, the following Maclaurin series  (with interval of convergence) are important:

❼ tan − 1 x = ( − , [ − 1, 1]

❼ ln(1 + x) = ( − , ( − 1, 1]

❼ ex  = , ( −∞ , ∞)

❼ sin x = ( 2(1) , ( −∞ , ∞)

❼ cos x = ( − , ( −∞ , ∞)

❼ sinh x = , ( −∞ , ∞)

❼ (1 + x)r   = (  )n(r) xn , ( − 1, 1) where the binomial ❼ cosh x = , ( −∞ , ∞)

coefficients are (n(r)) = r(r− −n+1)

If you need to construct a Maclaurin series of a function and some of the above functions are included, it is almost always easier to manipulate the Maclaurin series instead of constructing the series by scratch.

Approximating sums of series

In addition to finding whether sums of series converge or not, we also were able to find approximations to the error. There were 3 basic approximations to the error given by the Integral test, Alternating Series test, and the Taylor Series.

1. If 又 ak  is convergent with sum s and f(k) = ak  where f is a continuous, positive, and decreasing function for ∞

x ≥ n, then the remainder Rn  = s − sn  = 又 ak  satisfies the inequality

\n f(x)dx ≤ Rn ≤ \n∞ f(x)dx

2. If {ak } is a positive decreasing sequence with a limit of 0, then 又(− 1)k ak  is convergent with sum s and the

remainder Rn  = s − sn  = 又 ( − 1)k ak  satisfies the inequality

|Rn | < an+1

3. Taylor’s Inequality: If Tn (x) = (x − c)k  is the nth Taylor polynomial of f(x) centered at c, then the remainder Rn (x) = f(x) − Tn (x) satisfies the inequality

|Rn (x)| ≤ |x − c|n+1

on the interval where |f(n+1)(x)| < M .

We use this information, when applicable, to find maximum errors when approximating a function by a Taylor polynomial as well.

Questions

Try to study the review notes and memorize any relevant equations before trying to work these equations. If you cannot solve a problem without the book or notes, you will not be able to solve that problem on the exam.

Determine whether each sequence in 1 to 4 is con- vergent. State what it converges to, if applicable. Is the sequence increasing or decreasing? Is the sequence bounded?

1. an  =

2. an  = cos(nπ/2)

n sin n

3. an  =

1 + n2 .

5. Find the value that the sequence given by a1  = 1, an = converges to.

Determine whether the series in problems 6 to 8 is convergent or divergent. If it is con- vergent, find its sum.

6.工(∞)

∞

k=2

8. 工 n=0

Determine whether each series in question 9 to 17 converges or diverges. State any con- vergence/divergence tests you use.

∞

9. 工 n2 e −n n=1

10. ( )2

11.工(∞)

k=1

13. 工

14. 习

15. 习

16. 习

17. 习

18. Show that is an upper bound on the error of ∞ 1

习 if the sum is approximated by the first

19. Approximate the sum of

by summing the first 10 terms.  Find a bound on the error of your approximation.

For problems 20 through 25, determine whether the series is absolutely convergent, conditionally convergent, or divergent.

23. n1(n)        24. ( − 1)nn)2 25. ( − 1)n

26. Show that is an upper bound on the error of ∞ 1

习 if the sum is approximated by the first

∞

27. Suppose the power series 工 an (x+1)n has a radius

n=2

of convergence R = 5. List all possible intervals of convergence.

28. Find  the  radius  and  interval  of  convergence  of

工(∞) (x − 1)n

n=1

29. Find  the  radius  and  interval  of  convergence  of

工(∞) ( −4)n (x − 2)n

n=1

30. Find  the  radius  and  interval  of  convergence  of工(∞)

n=1

31. Find  the  radius  and  interval  of  convergence  of

32. Find  the  radius  and  interval  of  convergence  of工(∞)

n=1

33. Find a power series representation in powers of x for the function

f(x) =

with interval of convergence.

34. Find  a  power  series  representation  in  powers  of

(x − 1) for the function f(x) = 1北 and give the

35. Find  a  power  series  representation  in  powers  of (x − 1) for ln(1 + x).

36. What is the power series representation of (1 北(2))2 ?

37. Find the Maclaurin series for f(x) = ln(2 − x) from

of convergence.

38. Find a Taylor series for f(x) = cos(πx) centered at x  =  1.  Prove that the series you find represents cos(πx) for all x.

39. Use multiplication to find the first 4 terms of the Maclaurin series for f(x) = ex cosh(2x).

40. Use division to find the first 3 terms of the Maclau- rin series for g(x) = .

41. Use the power series of to estimate cor- rect to the nearest 0.001.  Justify that the error is less than 0.001 using the Alternating Series Estima- tion Theory or Taylor’s Inequality.

42. Find the sum:

(a) (^3)2

∞

(b) 对 2n(3)n!

n=0

(c)  对(∞) (

n=0

(d) + + + + ....

43. Find the Taylor polynomial T3 (x) for the function f(x) = arcsinx, at a = 0.

44. Approximate f by a Taylor polynomial with degree n  at  the  number  a.    And  use  Taylor’s  Inequal- ity to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x lies in the given interval.       (a) f(x) = ^3x,    a = 8,    n = 2,    7 ≤ x ≤ 9         (b) f(x) = x sin x,    a = 0,    n = 4,    − 1 ≤ x ≤ 1

45. Find the Taylor polynomial T3 (x) for the function f(x) = cosx at the number a = π/2. And use it to estimate cos800  correct to five decimal places.

46. A car is moving with speed 20m/s and acceleration 2m/s2   at a given instant.  Using a second-degree Taylor polynomial, estimate how far the car moves in the next second.  Would it be reasonable to use this polynomial to estimate distance traveled during the next minute?

47. Show that Tn  and f have the same derivatives at a up to order n.

Answers

converges to 0, decreasing, bounded

diverges, not increasing or decreasing, bounded.

converges  to   0,   not   increasing  or   decreasing, bounded.

diverges, increasing, bounded below.

− 1 +^5

diverges by the Divergence Test

Converges to 3/2 (Telescoping sum)

Converges to 15/4 (geometric series)

Use the integral test

\1 ∞ x2 e −x dx =

Therefore it converges by the integral test

Use the integral test

\1 ∞ ( )2 dx = 2

Therefore it converges by the integral test.

Use the integral test

\1 ∞ dx =

Therefore it converges by the integral test

\1 ∞ dx =

Therefore it converges by the integral test.

We have to be careful here since the function is not

defined at k = 1. By a change of variables, k = n+1

∞ ∞

Therefore it diverges by the integral test.

< = 2( )n  and之 2()n  converges (ge- ometric r = ). Thus  之(∞) converges by Com-

n=1

parison Test.

15. nl =  1 and 之 diverges. Hence 之 diverges  by  Limit

n=0

Comparison Test.

∞

16. > .  Thus 之 diverges by Comparison

n=1

Test.

17. < < and 之 converges (p-series ).

∞

Thus 之 converges by Comparison Test.

n=1

18. Since < it is sufficient to show that is a ∞ ∞

bound on the sum 之 . Then R2 ≤ 1 dx = .

n=3                                          2

19. 之 = 2.747299717. Note that

\10(∞) dx = .105,    \11(∞) dx = 0.09504132231.

Thus,      the &nbs