Ximpledu

Taylor Series

See how to write ex, sin x, cos x in taylor series.
3 examples and their solutions.

Taylor Series

Formula

f(x) = f(a)0!(x - a)0 + f'(a)1!(x - a)1 + f''(a)2!(x - a)2 + f'''(a)3!(x - a)3 + ... + f(n)(a)n!(x - a)n + ...
= n = 0f(n)(a)n!(x - a)n
A Taylor series is an approximation of y = f(x) near x = a.
It is useful because
you can approximate the value of a non-polynomial function
(trigonometric, exponential, logarithmic functions, etc)
by changing it to a polynomial function.
(Taylor series: Polynomial)
This is how a calculator finds the value
of a non-polynomial function.
f(n): nth derivative

Formula: Maclaurin Series

f(x) = f(0)0!x0 + f'(0)1!x1 + f''(0)2!x2 + f'''(0)3!x3 + ... + f(n)(0)n!xn + ...
= n = 0f(n)(0)n!xn
A Maclaurin Series is the special case of a Taylor series
near x = 0.

Example

Taylor series of ex near x = 0?
Solution

Example

Taylor series of sin x near x = 0?
Solution

Example

Taylor series of cos x near x = 0?
Solution