Taylor polynomial approximation

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Suppose that the function F:R->R has derivatives of all orders and that:

F"(x) - F'(x) - F(x) = 0 for all x
F(0)=1 and F'(0)=1

Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.

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Solution Preview

Please see the attached doc file .

First let's assume that:

then:

as f(0)=1 we conclude that a0=1 and as f'(0)=1 we see that a1=1. We also know that f''-f'-f=0 and if we plug the above expressions into this equation we have:

or we can unify the powers of x's in all ...

Solution Summary

This shows how to find a recursive formula for the coefficients of the nth Taylor polynomial for a given situation.

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