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 So, Thus, 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’s formula. So, Thus,