Maclairins theorem

Question 1

Maclairin’s theorem equation

Answer 1

f(x) = f(0) + f ‘ (0)/1! x + f‘’(0)/2!x^2 +f’’’(0)/3! *x^3…..

Question 2

Conditions for maclaurins theorem:

Answer 2

Each function must be differentiable up to the nth derivative.
Each derivative can be evaluated for x=0

Question 3

Maclaurin series for e^x=

Answer 3

e^x = 1 + x +1/2x^2 + 1/6x^3 + 1/24*x^4……

Question 4

Maclaurin series for ln(1+x)

Answer 4

ln(1+x) = x -1/2x^2 + 1/3x^3 - 1/4*x^4+…

Question 5

Maclaurin series for sin(x)

Answer 5

Sin(x) = x-1/3!x^3 + 1/5!x^5 -1/7!x^7 +…..

Question 6

Maclaurin series for cos(x)

Answer 6

1-1/2!x^2 +1/4!x^4 - 1/6!*x^6…..

Question 7

Maclaurins series for tan^-1(x)

Answer 7

tan^-1(x) = x - 1/3x^3 +1/5x^5-1/7*x^7

Question 8

Combining expressions when one multiplies the other.

Answer 8

First write out full maclaurins theorem for expressions up to max power, substituting in 3x for example if necessary.
The write combined with one brackets series and another bracket series. Expand brackets ignoring answers that will give too high powers. Combine like terms.

Question 9

Combining expressions when one is to the power of the other (e^func(x)

Answer 9

Write out each series of functions separately to max power, subbing in if required.
Then substitute in func x for x in e^x equation. Multiply out bearing in mind highest power, combine like terms.