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The Maclaurin Formula:
Notes
(i) f (x) is the function given.
(ii) f' (0) = the value of the first derivative at 0
f'' (0) = the value of the second derivative at 0 etc..
2. The Functions
The functions to be expanded are: ex, sin x, cos x, ln (1 + x), (1 + x)p, tan-1 x
Notes:
(i) The approximations for ex, sin x and cos x are valid (convergent) for all values according to the Ratio Test.
(ii) The approximations for
1(1 + x)p is valid for -1 < x < 1
tan-1 x is valid iff |x|
1 (iii) The Mac for tan-1 x must be obtained in a roundabout fashion. For this reason we will refer to it as the OMO (odd man out).
(iv) Some Macs can be deduced from others.
(v) Be able to write down the general term un for each series.
3. The Exponential Mac
Example 1
Find the Mac for ex as far as x4.Deduce the Mac for (i) e2x (ii)
Find the general term un and test it for convergence.
Solution
Trick: If asked to the Mac for a function like
don’t. Do the Mac for ex and then replace x by f (x).4. The Trig Mac
Example 2
Find the Mac for sin x as far as x5.Deduce the Macs for (i) cos x (ii) cos 2x
(iii) sin2 x.
Find un for sin x and show it is convergent.
Solution
5. The Log Mac
Example 3
Find the Mac for (1 + x)-1 as far as x4.Deduce the Mac for (i) ln (1 + x) (ii) ln (1 - x) (iii)
Use the ratio test to show that the Mac for ln (1 + x) is convergent
for |x| < 1.
Solution
6. The Inverse Mac
Example 4
Find the Mac for (1 + x)-1 as far as x3.Deduce the Mac for (i)
as far as x6(ii) tan-1 x as far as x7
Show
Hence estimate p.
Solution
7. General
(i) ex, sin x and cos x are convergent for all x. So you can plonk in any value of x.
(ii) (1 + x)p and ln (1 + x) are only convergent for |x| < 1. So you can only plonk in values x such that |x| < 1.
(iii) tan-1 x is convergent for |x|
1.
