Remainder Estimation Theorem
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Remainder Estimation Theorem Section 9.3b
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Remainder Estimation Theorem. Section 9.3b. Remainder Estimation Theorem. In the last class, we proved the convergence to a Taylor s eries to its generating function (sin( x )), and yet we did n ot need to find any actual values for the derivatives of t he function!. - PowerPoint PPT Presentation
Transcript of Remainder Estimation Theorem
Remainder Estimation Theorem
Section 9.3b
Remainder Estimation TheoremIn the last class, we proved the convergence to a Taylorseries to its generating function (sin(x)), and yet we didnot need to find any actual values for the derivatives ofthe function!
Instead, we were able to find an upper bound on thederivatives, which was enough to ensure that theremainder converged to zero for all x. This is thefoundation of the new theorem…
Remainder Estimation TheoremIf there are positive constants M and r such that
for all t between a and x, then the 1 1n nf t Mr remainder in Taylor’s Theorem satisfies the nR x
inequality
11
1 !
nn
n
r x aR x M
n
If these conditions hold for every n and all the otherconditions of Taylor’s Theorem are satisfied by f, thenthe series converges to f (x).
Remainder Estimation TheoremIf there are positive constants M and r such that
for all t between a and x, then the 1 1n nf t Mr remainder in Taylor’s Theorem satisfies the nR x
inequality
11
1 !
nn
n
r x aR x M
n
(It doesn’t matter if M and r are huge, the importantthing is that they don’t get any more huge as n approaches infinity. This allows the factorial growth tooutstrip the power growth and thereby sweep theremainder to zero.)
Remainder Estimation TheoremUse the Remainder Estimation Theorem to prove that
0 !
kx
k
xek
for all real x.
By the theorem, we need to find M and r such that 1n tf t e is bounded by for t between 0 1nMr
and an arbitrary x…We know that the exponential function is increasing onany interval, so it reaches its maximum value at theright-hand endpoint. We can pick M to be thatmaximum value and simply let r = 1.
Remainder Estimation TheoremUse the Remainder Estimation Theorem to prove that
0 !
kx
k
xek
for all real x.
If the interval is [0, x], we let xM eIf the interval is [x, 0], we let 0 1M e
In either case, we have throughout the interval,and the Remainder Estimation Theorem guaranteesconvergence!
te M
Remainder Estimation TheoremThe approximation is usedwhen x is small. Use the Remainder Estimation Theoremto get a bound for the maximum error when .
2ln 1 2x x x
0.1x
ln 1f x x 22 2P x x x
We need a bound for : 2R x
Look at on [–0.1, 0.1]
33
21
f tt
It is strictly decreasing on the interval, achieving itsmaximum value at the left-hand endpoint, –0.1.
Remainder Estimation TheoremThe approximation is usedwhen x is small. Use the Remainder Estimation Theoremto get a bound for the maximum error when .
2ln 1 2x x x
0.1x
ln 1f x x 22 2P x x x
We can bound by 3f t
And we can let r = 1. 32
1 0.1M
2000729
Remainder Estimation TheoremThe approximation is usedwhen x is small. Use the Remainder Estimation Theoremto get a bound for the maximum error when .
2ln 1 2x x x
0.1x
ln 1f x x 22 2P x x x
Conclusion from the Remainder Estimation Theorem:
3
22000729 3!
xR x
30.12000729 3!
44.6 10
We can even graph this remainder term: 2
2 ln 1 2R x x x x
Remainder Estimation TheoremThe approximation is usedwhen x is small. Use the Remainder Estimation Theoremto get a bound for the maximum error when .
2ln 1 2x x x
0.1x
Graph in [–0.12, 0.12] by [–0.0005, 0.0005] 2
2 ln 1 2R x x x x
2 0.1R 43.605 10 2 0.1R 43.102 10
So the absolute error on the interval: 43.605 10
Which is indeed less than the bound: 44.6 10
Calculate:
Euler’s FormulaWe have now seen that sine, cosine, and the exponentialfunctions equal their respective Maclaurin series for allreal numbers x. Now, let’s see what happens when weassume that this is also true for all complex numbers…
1i
etc.
Recall the powers of :1i i2 1i 3i i4 1i 5i i
Euler’s FormulaAssume that the exponential, cosine, and sine functionsequal their Maclaurin series (as in the table in Section9.2) for complex numbers as well as for real numbers.
ixe1. Find the Maclaurin series for
2 3
12! 3! !
nix ix ix ixe ix
n
2 3 4 5
12! 3! 4! 5! !
nnx x x x xix i i in
Euler’s Formula2. Use the result of part 1 and the Maclaurin series forcos(x) and sin(x) to prove that cos sinixe x i x
This equation is known as Euler’s formula.
2 3 4 5
12! 3! 4! 5! !
nnix x x x x xe ix i i in
2 4 6 2
1 12! 4! 6! 2 !
nnx x x xn
3 5 7 2 1
13! 5! 7! 2 1 !
nnx x x xi xn
cos sinx i x
Euler’s Formula3. Use Euler’s formula to prove that . Thisbeautiful equation, which brings together some of themost celebrated numbers in mathematics in such astunningly unexpected way, is also widely known asEuler’s formula.
1 0ie
cos sinixe x i x
cos sinie i 1 0 1 So,
1 0ie Thus,
