Calculus I – Math 104

The end is near!

Series approximations for functions, integrals etc..

We've been associating series with functions and using them to evaluate limits, integrals and such.

We have not thought too much about how good the approximations are. For serious applications, it is important to do that.

Questions you can ask--

1. If I use only the first three terms of the series, how big is the error?

2. How many terms do I need to get the error smaller than 0.0001?

To get error estimates:

Use a generalization of the Mean Value Theorem for derivatives

Derivative MVT approach:

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If you know...

If you know that the absolute value of the derivative is always less than M, then you know that

| f(x) - f(0) | < M |x|

The derivative form of the error estimate for series is a generalization of this.

Lagrange's form of the remainder:

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Lagrange...

Lagrange's form of the remainder looks a lot like what would be the next term of the series, except the n+1 st derivative is evaluated at an unknown point between 0 and x, rather than at 0:

So if we know bounds on the n+1st derivative of f, we can bound the error in the approximation.

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Example: The series for sin(x) was:

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5th derivative

For f(x) = sin(x), the fifth derivative is f '''''(x) = cos(x). And we know that |cos(t)| < 1 for all t between 0 and x. We can conclude from this that:

So for instance, we can conclude that the approximation sin(1) = 1 - 1/6 = 5/6 is accurate to within 1/5! = 1/120 -- i.e., to two decimal places.

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Another application...Another application of Lagrange's form of the

remainder is to prove that the series of a function actually converges to the function. For example, for the series for sin(x), we have (since all the derivatives of sin(x) are always less than or equal to 1 in absolute value):

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Shifting the origin -- Taylor vs Maclaurin

So far, we've been writing all of our series as infinite polynomials and using values of the function f(x) and its derivatives evaluated at x=0. It is possible to change one's point of view and use values of the function and derivatives at other points.

As an example, we’ll return to the geometric series

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Taylor series

By taking derivatives of the function g(x) = -1/x and evaluating them at x=-1, we will discover that the expansion of g(x) we have found is the Taylor series for g(x) expanded around -1:

g(x) = g(-1) + g '(-1) (x+1) + g ''(-1) + ....

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Note:

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Maclaurin

Series expansions around points other than zero are useful when trying to approximate function values for x far from zero, but close to a different point where much is known about the function.

But note that by defining a new function g(x) = f(x+a), you can use Maclaurin expansions for g instead of general Taylor expansions for f.

Binomial series

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Fibonacci numbers

Everyone is probably familiar with the famous sequence of Fibonacci numbers. The idea is that you start with 1 (pair of) rabbit(s) the zeroth month. The first month you still have 1 pair. But then in the second month you have 1+1 = 2 pairs, the third you have 1 + 2 = 3 pairs, the fourth, 2 + 3 = 5 pairs, etc... The pattern is that if you have a pairs in the nth month, and a pairs in the n+1st month, then you will have pairs in the n+2nd month.

The first several terms of the sequence are thus:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc...

Is there a general formula for a ? n

n n+1

n+1na + a

Generating functionsThis is a common problem in many parts of

mathematics and science. And a powerful method for solving such problems involves series -- which in this case are called generating functions for their sequences.

For the Fibonacci numbers, we will simply define a function f(x) via the series:

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Recurrence relation

To do this, we'll use the fact that multiplication by x "shifts" the series for f(x) as follows:

Now, subtract the second two from the first -- almost everything will cancel because of the recurrence relation!

...)(f

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