Question

whats the maclaurin series of fx = (1-cos(x^7)/ x^3)....? need some help ...

Answer 1

thats pretty easy

Answer 2

the denominator is already in powers of x, leave that alone

Answer 3

then just need to know the general maclaurin series for cos(x)

Answer 4

\[\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - .... \]

Answer 5

replace the x with x^7

Answer 6

\[(1- \cos(x^7) ) = (1 - ( 1- \frac{(x^7)^2}{2!} + \frac{(x^7)^4}{4!} - ..... ) )\]

Answer 7

\[(1-\cos(x^7) ) = \frac{x^{14}}{2!} - \frac{x^{28}}{4!} + .... \]

Answer 8

\[\frac{1-\cos(x^7)}{x^3} = \frac{x^{11}}{2!} - \frac{x^{25}}{4!} + .... \]

Answer 9

etc etc, you can get a general form if you really want,

Answer 10

how do you know when to stop the serioes? the general form?

Answer 11

it doesnt stop obviously

Answer 12

you just write down the first two or three terms, then write .... after wards

Answer 13

so they know it continues , its an infinite series

Answer 14

so if n=1 you just give em the first term? because they want the interval of convergence also