Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that displaystyle{R}{n}{left({x}right)}rightarrow{0}.] displaystyle f{{left({x}right)}}={e}-{5}{x}

Zoe Oneal 2021-02-21 Answered
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x)0.]
f(x)=e5x
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Expert Answer

Jaylen Fountain
Answered 2021-02-22 Author has 170 answers
Since we are assuming that ff has a power series expansion, we may apply the Mclauren series expansion general formula, which states that the power series expansion of ff is given by:
f(x)=n=0fn(0)xnn!
We simply find the general formula for the nth derivative and plug in x=0 to it.
We go by the pattern first:
f=e5x=(1)050e5x
f=5e5x=(1)151e5x
f=52e5x=(1)252e5x
...
Generally, we can see the pattern shows that the general formula is:
fn(x)=(1)n5ne5x
Therefore:
fn(0)=(1)n5n=(5)n
Thus, our Mclaurin series expansion is:
f(x)=e5x=n=0(5)nxnn!
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