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. What i mean by general taylor polynomail is for example
1/(1-x)=1+x+x^2+x^3...+x^n. What is need is after ....
I would write it as
(1+x^4)^-1 and apply the general binomial theorem to get
= 1^-1 + (-1)(1^-2/1!(x^4) + (-1)(-2)(1^-3)/2! (x^4)^2 + (-1)(-2)(-3)(1^-3)/3! (x^4)^3 + ...
= 1 - x^4 + x^8 - x^12 + x^16 - x^20 + ...
for -1 < x < +1
I tested for x = .25 and my margin of error was 4.17x10^-10 using the above 6 terms
To represent the pattern 1 - x^4 + x^8 - x^16 + ... in a general Taylor polynomial, you can use the concept of geometric series.
A geometric series has the form:
a + ar + ar^2 + ar^3 + ...
In this case, the pattern 1 - x^4 + x^8 - x^16 + ... can be expressed as a geometric series, where a = 1 and r = -x^4.
Now, using the formula for the sum of an infinite geometric series, which is:
S = a / (1 - r)
We can plug in the values a = 1 and r = -x^4:
S = 1 / (1 + x^4)
This represents the infinite sum of the series 1 - x^4 + x^8 - x^16 + ... in a general Taylor polynomial form, where 'S' represents the value of the series.