Maclaurin Series PDF

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Question & Solutions to solve Maclaurin Series...

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Maclaurin Series Question 1 Determine the maclaurin series for f(x) = cos(7x)

Answer 1 We are working with the cosine function and want the maclaurin series (same as the Taylor series centered about x=0). The recommended Calculus book “Early Transcendentals” comprises of the following table

Considering that we know the Maclaurin expansion for cos x (highlighted above), we simply substitute 7x instead of x in this expression. Thus we get cos(7x) = =

(-1)n (7x)2n / (2n)! (-1)n 49n x2n / (2n)!

Question 2 Determine the maclaurin series for f(x) = x8 e3x

Answer 2 We are working with the exponential function (ex) and want the maclaurin series (same as the Taylor series centered about x=0). The recommended Calculus book “Early Transcendentals” comprises of the following table

Here, we only convert the exponential using the above expression (highlighted) and simply substitute 3x instead of x. We also leave the x8 in front of the series and incorporate the same later. x8 e3x = x8 =

(3x)n / n! 3n xn+8 / n!

Note that the basic rules of a series state that we don’t want anything out in front of the series and we want a single x with a single exponent on it.

Question 3 Evaluate the limit using the maclaurin expansion sin(x) – x / x3 (Please note that since this is the 0/0 form one could use L’Hospital’s rule but the question clearly states use maclaurin expansion)

Answer 3 We are working with the sine function and want the maclaurin series (same as the Taylor series centered about x=0). The recommended Calculus book “Early Transcendentals” comprises of the following table

To evaluate the limit we use the maclaurin expansion for sinx and then evaluate sinx -x.

sinx – x = (

)–x =

Now dividing each term by x3, we get the following sin(x) – x / x3 =

=

/ x3

=

- 1 / 3! = -1/6...