Recipe

Build a full Taylor or Maclaurin series

Use when: the pattern in a function's derivatives at the center is clean enough to write a general n-th term, not just a fixed finite-degree polynomial

Goal: the infinite series sum f^(n)(a)/n! (x-a)^n, in closed sigma-notation form

Taught in Taylor and Maclaurin series · extends Build a Taylor polynomial centered at a

1. 1

   Differentiate f repeatedly at the center until the pattern in the derivatives (their sign, and which ones vanish) becomes clear.

   Example

   f(x)=e^{2x}, centered at a=0 (a Maclaurin series): f^{(n)}(x)=2^ne^{2x} for every n, so f^{(n)}(0)=2^n.

   Why: an infinite series, unlike one finite-degree polynomial, needs a clean pattern so every coefficient is computable at once, not one at a time.

2. 2

   Express \dfrac{f^{(n)}(a)}{n!} as a single formula in n, valid for every n from 0 to infinity.

   Example

   \frac{f^{(n)}(0)}{n!} = \frac{2^n}{n!}
   • the center is a=0 — the series is specifically called a Maclaurin series, a special case of the general Taylor series.
   • the center is any other value a — the series is a Taylor series centered at a, using (x-a)^n instead of x^n.

   Why: a single closed-form coefficient formula is what turns Step 1's derivative pattern into a genuine infinite series instead of a list of separately-computed numbers.

3. 3

   Write \displaystyle\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n, substituting the coefficient formula from Step 2.

   Example

   \boxed{e^{2x} = \sum_{n=0}^\infty\frac{2^n}{n!}x^n}

   Why: this single sigma-notation expression stands for every partial sum T_n(x) from the finite-polynomial recipe, all at once, as n\to\infty.

4. 4

   Expand the series' first few terms by hand and confirm they match the Taylor polynomial the earlier recipe would produce at that same degree.

   Example

   \sum_{n=0}^\infty\frac{2^n}{n!}x^n = 1+2x+2x^2+\frac43x^3+\cdots

   matching T_2(x)=1+2x+2x^2 from the degree-2 polynomial built the same way.

   Why: writing the series down only ever guarantees a formal expression. Nothing here proves it converges to f(x) away from the center; that's what a later section's remainder bound establishes.