Common Maclaurin Series and Radii of Convergence

A collection of flashcards detailing the Maclaurin series expansions and their respective radii of convergence for fundamental mathematical functions.

Maclaurin series for $\frac{1}{1-x}$

$$\sum_{n=0}^{\infty} x^n$$

Radius of convergence ($R$) for $\frac{1}{1-x}$

$$R = 1$$

Maclaurin series for $e^x$

$$\sum_{n=0}^{\infty} \frac{1}{n!} x^n$$

Radius of convergence ($R$) for $e^x$

$$R = \infty$$

Maclaurin series for $\ln(1+x)$

$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^n$$

Radius of convergence ($R$) for $\ln(1+x)$

$$R = 1$$

Radius of convergence ($R$) for $(1+x)^k$

$$R = 1$$

Maclaurin series for $\sin(x)$

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

Radius of convergence ($R$) for $\sin(x)$

$$R = \infty$$

Maclaurin series for $\cos(x)$

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

Radius of convergence ($R$) for $\cos(x)$

$$R = \infty$$