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Calculus II: Series Cheat Sheet

Calculus II: Series Cheat Sheet










Calculus II: Series Cheat Sheet




Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point a.


$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n $$


Expanded, this looks like:


$$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots $$


Here, f(n)(a) is the nth derivative of f evaluated at the point a.






Maclaurin Series

A Maclaurin series is a special case of the Taylor series, centered at a = 0. This is the most common type of power series expansion.


$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n $$


Expanded, this looks like:


$$ f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots $$







Common Maclaurin Series (Your Building Blocks)

Memorizing or having these handy is crucial. Most problems involve manipulating these known series rather than deriving them from scratch.





Function

Series Expansion

Interval of Convergence






$$ \frac{1}{1-x} $$

$$ \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots $$

$$ (-1, 1) $$




$$ e^x $$

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

$$ (-\infty, \infty) $$




$$ \sin(x) $$

$$ \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots $$

$$ (-\infty, \infty) $$




$$ \cos(x) $$

$$ \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots $$

$$ (-\infty, \infty) $$




$$ \ln(1+x) $$

$$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots $$

$$ (-1, 1] $$




$$ \arctan(x) $$

$$ \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots $$

$$ [-1, 1] $$




$$ (1+x)^k $$ (Binomial Series)

$$ 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \cdots $$

$$ (-1, 1) $$












Techniques for Finding New Series

You can use the known series above to find new ones through simple algebraic manipulation.



Substitution: Replace x with another function.
Example: To find the series for e^{-x^3}, substitute u = -x^3 into t…