This is an example of an infinite series. 1 1 Start with a square one unit by one unit: This series converges (approaches a limiting value.) Many series do not converge: 9.1 Power Series

In an infinite series: a 1, a 2,… are terms of the series. a n is the n th term. Partial sums: n th partial sum If S n has a limit as, then the series converges, otherwise it diverges. 9.1 Power Series

Geometric Series: In a geometric series, each term is found by multiplying the preceding term by the same number, r. This converges to if, and diverges if. is the interval of convergence. 9.1 Power Series

a r

a r

A power series is in this form: or The coefficients c 0, c 1, c 2 … are constants. The center “ a ” is also a constant. (The first series would be centered at the origin if you graphed it. The second series would be shifted left or right. “ a ” is the new center.) 9.1 Power Series

The partial sum of a geometric series is: If then If and we let, then: 9.1 Power Series

We could generate this same series for with polynomial long division: 9.1 Power Series

Once we have a series that we know, we can find a new series by doing the same thing to the left and right hand sides of the equation. This is a geometric series where r=-x. To find a series for multiply both sides by x. 9.1 Power Series

Given:find: So: We differentiated term by term. 9.1 Power Series

Given:find: hmm?

The previous examples of infinite series approximated simple functions such as or. This series would allow us to calculate a transcendental function to as much accuracy as we like using only pencil and paper! 9.1 Power Series

Brook Taylor Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. 9.2 Taylor Series

Suppose we wanted to find a fourth degree polynomial of the form: atthat approximates the behavior of If we make, and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation. 9.2 Taylor Series

If we plot both functions, we see that near zero the functions match very well! 9.2 Taylor Series

This pattern occurs no matter what the original function was! Our polynomial: has the form: or: 9.2 Taylor Series

Maclaurin Series: (generated by f at ) 9.2 Taylor Series

If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series: Taylor Series: (generated by f at ) 9.2 Taylor Series

The more terms we add, the better our approximation. 9.2 Taylor Series

Rather than start from scratch, we can use the function that we already know: 9.2 Taylor Series

example: 9.2 Taylor Series

There are some Maclaurin series that occur often enough that they should be memorized. They are on pg Taylor Series

When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 terms. 9.2 Taylor Series

It is a 4th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power. The 3rd order polynomial for is, but it is degree 2. The x 3 term drops out when using the third derivative. This is also the 2nd order polynomial. 9.2 Taylor Series

List the function and its derivatives. 9.2 Taylor Series

Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x. 9.2 Taylor Series

We could generate this same series for with polynomial long division: 9.2 Taylor Series

This is a geometric series with a = 1 and r = -x. 9.2 Taylor Series

We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly. We will find other uses for these series, as well. They do help to explain where the formula for the sum of an infinite geometric comes from. A more impressive use of Taylor series is to evaluate transcendental functions. 9.2 Taylor Series

Both sides are even functions. Cos (0) = 1 for both sides. 9.2 Taylor Series

Both sides are odd functions. Sin (0) = 0 for both sides. 9.2 Taylor Series

If we start with this function: and substitute for, we get: This is a geometric series with a = 1 and r = -x Taylor Series

If we integrate both sides: This looks the same as the series for sin ( x ), but without the factorials. 9.2 Taylor Series

Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use the calculator or computer to calculate directly.) An estimate is only useful if we have an idea of how accurate the estimate is. When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is. 9.3 Taylor’s Theorem

For a geometric series, this is easy: Use to approximate over. Since the truncated part of the series is:, the truncation error is, which is. When you “truncate” a number, you drop off the end. Of course this is also trivial, because we have a formula that allows us to calculate the sum of a geometric series directly. 9.3 Taylor’s Theorem

Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I : 9.3 Taylor’s Theorem

Lagrange Form of the Remainder Remainder after partial sum S n where c is between a and x. 9.3 Taylor’s Theorem

This is also called the remainder of order n or the error term. Note that this looks just like the next term in the series, but “ a ” has been replaced by the number “ c ” in. This seems kind of vague, since we don’t know the value of c, but we can sometimes find a maximum value for. 9.3 Taylor’s Theorem

We will call this the Remainder Estimation Theorem. Lagrange Form of the Remainder Remainder Estimation Theorem Note that this is not the formula that is in our book. It is from another textbook. If M is the maximum value of on the interval between a and x, then: 9.3 Taylor’s Theorem

Prove that, which is the Taylor series for sin x, converges for all real x. 9.3 Taylor’s Theorem

Since the maximum value of sin x or any of it’s derivatives is 1, for all real x, M = 1. so the series converges. Remainder Estimation Theorem 9.3 Taylor’s Theorem

Find the Lagrange Error Bound when is used to approximate and. Remainder after 2nd order term On the interval, decreases, so its maximum value occurs at the left end-point. 9.3 Taylor’s Theorem

Find the Lagrange Error Bound when is used to approximate and. On the interval, decreases, so its maximum value occurs at the left end-point. Remainder Estimation Theorem Lagrange Error Bound error Error is less than error bound. 9.3 Taylor’s Theorem

We have saved the best for last! 9.3 Taylor Series

An amazing use for infinite series: Substitute xi for x. 9.3 Taylor Series

Factor out the i terms. 9.3 Taylor Series

This is the series for cosine. This is the series for sine. Let This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated. 9.3 Taylor Series

Convergence The series that are of the most interest to us are those that converge. Today we will consider the question: “Does this series converge, and if so, for what values of x does it converge?” 9.4 Radius of Convergence

The first requirement of convergence is that the terms must approach zero. n th term test for divergence diverges if fails to exist or is not zero. Note that this can prove that a series diverges, but can not prove that a series converges. 9.4 Radius of Convergence

If then grows without bound. If then As, eventually is larger than, therefore the numerator grows faster than the denominator. The series diverges. (except when x=0) 9.4 Radius of Convergence

1The series converges over some finite interval: (the interval of convergence). The series may or may not converge at the endpoints of the interval. There is a positive number R such that the series diverges for but converges for. 2 The series converges for every x. ( ) 3 The series converges for at and diverges everywhere else. ( ) The number R is the radius of convergence. 9.4 Radius of Convergence

This series converges. So this series must also converge. Direct Comparison Test For non-negative series: If every term of a series is less than the corresponding term of a convergent series, then both series converge. If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. So this series must also diverge. This series diverges. 9.4 Radius of Convergence

Ex. 3: Prove that converges for all real x. There are no negative terms: is the Taylor series for, which converges. larger denominator The original series converges. The direct comparison test only works when the terms are non-negative. 9.4 Radius of Convergence

Absolute Convergence If converges, then we say converges absolutely. The term “converges absolutely” means that the series formed by taking the absolute value of each term converges. Sometimes in the English language we use the word “absolutely” to mean “really” or “actually”. This is not the case here! 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.” 9.4 Radius of Convergence

We test for absolute convergence: Since, 9.4 Radius of Convergence

Since, converges to converges by the direct comparison test. Since converges absolutely, it converges. 9.4 Radius of Convergence

Ratio Technique We have learned that the partial sum of a geometric series is given by: where r = common ratio between terms When, the series converges. 9.4 Radius of Convergence

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. 9.4 Radius of Convergence

For, if then: if the series converges.if the series diverges.if the series may or may not converge. 9.4 Radius of Convergence

Ex: If we replace x with x- 1, we get: 9.4 Radius of Convergence

If the limit of the ratio between consecutive terms is less than one, then the series will converge. 9.4 Radius of Convergence

The interval of convergence is (0,2). The radius of convergence is 1. If the limit of the ratio between consecutive terms is less than one, then the series will converge. 9.4 Radius of Convergence

The interval of convergence is (2,8). The radius of convergence is. 9.4 Radius of Convergence

for all. Radius of convergence = 0. At, the series is, which converges to zero. Note: If r is infinite, then the series converges for all values of x. 9.4 Radius of Convergence

Another series for which it is easy to find the sum is the telescoping series. Telescoping Series converges to 9.4 Radius of Convergence by partial fractions

This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series. 9.5 Testing Convergence

The series converges if. The series diverges if. The test is inconclusive if. Nth Root Test: If is a series with positive terms and then: Note that the rules are the same as for the Ratio Test. 9.5 Testing Convergence

?

Indeterminate, so we use L’Hôpital’s Rule 9.5 Testing Convergence

example: it converges ? 9.5 Testing Convergence

another example: it diverges 9.5 Testing Convergence

Remember: The series converges if. The series diverges if. The test is inconclusive if. The Ratio Test: If is a series with positive terms and then: 9.5 Testing Convergence

This leads to: The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge. 9.5 Testing Convergence

Example 1: Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.) 9.5 Testing Convergence

p-series Test converges if, diverges if. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster. 9.5 Testing Convergence

the harmonic series: diverges. (It is a p-series with p=1.) It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series. 9.5 Testing Convergence

Limit Comparison Test If and for all ( N a positive integer) If, then both and converge or both diverge. If, then converges if converges.If, then diverges if diverges.

When n is large, the function behaves like: Since diverges, the series diverges. harmonic series 9.5 Testing Convergence

When n is large, the function behaves like: Since converges, the series converges. geometric series 9.5 Testing Convergence

1.each u n is positive; 2.u n > u n+1 for all n > N for some integer N (decreasing); 3.lim n →  u n ⃗ 0 Theorem 12 The Alternating Series Test The series converges if all three of the following conditions are satisfied:

Alternating Series example: This series converges (by the Alternating Series Test.) If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series Test This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent. 9.5 Testing Convergence

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 a good tool to remember, because it is easier than the LaGrange Error Bound. 9.5 Testing Convergence

series diverges Converges to a/(1-r) if |r|<1. Diverges if |r|>1 Converges if p>1 Diverges if p<1 non-negative terms and/or absolute convergence Does Σ |a n | converge? Apply Integral Test, Ratio Test or nth-root Test Original Series Converges Alternating Series Test Is Σa n = u 1 -u 2 +u 3 -… an alternating series Is there an integer N such that u N >u N-1 …? Converges if u n  0 Diverges if u n  0 nth-Term Test Is lim a n =0 no yes or maybe no no or maybe Geometric Series Test Is Σa n = a+ar+ar 2 + … ? yes p-Series Test Is series form