find maclaurin series.
f(x)=tanx
To find the Maclaurin series expansion of the function f(x) = tan(x), we can use the fact that the Maclaurin series is simply the Taylor series centered at x = 0.
The general formula for the Maclaurin series of a function f(x) is:
f(x) = f(0) + f'(0)x + f''(0)(x^2 / 2!) + f'''(0)(x^3 / 3!) + ...
For the function f(x) = tan(x), we will start by finding the first few derivatives:
f(x) = tan(x)
f'(x) = sec^2(x)
f''(x) = 2sec^2(x)tan(x)
f'''(x) = 2sec^2(x)sec^2(x) + 4sec^4(x)tan(x)
Now, we need to evaluate these derivatives at x = 0:
f(0) = tan(0) = 0
f'(0) = sec^2(0) = 1
f''(0) = 2sec^2(0)tan(0) = 0
f'''(0) = 2sec^4(0) + 4sec^4(0)tan(0) = 2
Now we can substitute these values into the general formula for the Maclaurin series:
tan(x) ≈ 0 + 1x + 0(x^2 / 2!) + 2(x^3 / 3!) + ...
Simplifying further:
tan(x) ≈ x + (2/3)x^3 + (4/15)x^5 + ...
So, the Maclaurin series expansion of tan(x) is:
tan(x) ≈ x + (2/3)x^3 + (4/15)x^5 + ...