- #1

danago

Gold Member

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**Show that the taylor series generated by [tex]f(x)=e^{x}[/tex] about x=0 converges to f(x) for ever real value of x.**

Taylors theorem states that:

[tex]

f(b) = P_{n} + \frac{e^c}{(n+1)!} b^{n+1}

[/tex]

where P_n is the taylor polynomial of order 'n' and the following term is the error term and c is some value between 0 and b. If the error term approaches 0 as n approaches infinity, then the series converges to f(x).

Does this mean that to answer the question all i need to do is show that the error term approaches zero as n gets large? If so, how would i do so?

Thanks in advance for the help,

Dan.