how do i represent 1/(1+x^4) in general taylor polynomial. I know the pattern is 1-x^4+x^8-x^16..... I don't know how to represent this pattern in variables equation form

To represent the pattern in the general Taylor polynomial form, we can start by noticing that the pattern alternates between positive and negative powers of x. This suggests that we can introduce the (-1)^n factor to achieve the alternation.

Next, we observe that the exponent of x increases by a power of 4 with each term. This can be expressed as n*4, where n is the index of the term.

Therefore, we can write the general term of the Taylor polynomial of 1/(1+x^4) as follows:

(-1)^n * x^(n*4)

Where n = 0, 1, 2, 3, …

This expression represents the alternating pattern of positive and negative powers of x with the appropriate exponent.