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Remainder Estimation Theorem

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Remainder Estimation Theorem

Section 9.3b

Remainder Estimation Theorem

In the last class, we proved the convergence to a Taylor

series to its generating function (sin(x)), and yet we did

not need to find any actual values for the derivatives of

the function!

Instead, we were able to find an upper bound on the

derivatives, which was enough to ensure that the

remainder converged to zero for all x. This is the

foundation of the new theorem…

Remainder Estimation Theorem

If there are positive constants M and r such that

for all t between a and x, then the

remainder in Taylor’s Theorem satisfies the

inequality

If these conditions hold for every n and all the other

conditions of Taylor’s Theorem are satisfied by f, then

the series converges to f (x).

Remainder Estimation Theorem

If there are positive constants M and r such that

for all t between a and x, then the

remainder in Taylor’s Theorem satisfies the

inequality

(It doesn’t matter if M and r are huge, the important

thing is that they don’t get any more huge as n

approaches infinity. This allows the factorial growth to

outstrip the power growth and thereby sweep the

remainder to zero.)

Remainder Estimation Theorem

Use the Remainder Estimation Theorem to prove that

for all real x.

By the theorem, we need to find M and r such that

is bounded by for t between 0

and an arbitrary x…

We know that the exponential function is increasing on

any interval, so it reaches its maximum value at the

right-hand endpoint. We can pick M to be that

maximum value and simply let r = 1.

Remainder Estimation Theorem

Use the Remainder Estimation Theorem to prove that

for all real x.

If the interval is [0, x], we let

If the interval is [x, 0], we let

In either case, we have throughout the interval,

and the Remainder Estimation Theorem guarantees

convergence!

Remainder Estimation Theorem

The approximation is used

when x is small. Use the Remainder Estimation Theorem

to get a bound for the maximum error when .

We need a bound for :

Look at on [–0.1, 0.1]

It is strictly decreasing on the interval, achieving its

maximum value at the left-hand endpoint, –0.1.

Remainder Estimation Theorem

The approximation is used

when x is small. Use the Remainder Estimation Theorem

to get a bound for the maximum error when .

We can bound by

And we can let r = 1.

Remainder Estimation Theorem

The approximation is used

when x is small. Use the Remainder Estimation Theorem

to get a bound for the maximum error when .

Conclusion from the Remainder Estimation Theorem:

We can even graph this remainder term:

Remainder Estimation Theorem

The approximation is used

when x is small. Use the Remainder Estimation Theorem

to get a bound for the maximum error when .

Graph in [–0.12, 0.12] by [–0.0005, 0.0005]

Calculate:

So the absolute error on the interval:

Which is indeed less than the bound:

Euler’s Formula

We have now seen that sine, cosine, and the exponential

functions equal their respective Maclaurin series for all

real numbers x. Now, let’s see what happens when we

assume that this is also true for all complex numbers…

Recall the powers of :

etc.

Euler’s Formula

Assume that the exponential, cosine, and sine functions

equal their Maclaurin series (as in the table in Section

9.2) for complex numbers as well as for real numbers.

1. Find the Maclaurin series for

Euler’s Formula

2. Use the result of part 1 and the Maclaurin series for

cos(x) and sin(x) to prove that

This equation is known as Euler’s formula.

Euler’s Formula

3. Use Euler’s formula to prove that . This

beautiful equation, which brings together some of the

most celebrated numbers in mathematics in such a

stunningly unexpected way, is also widely known as

Euler’sformula.

So,

Thus,