Maclaurin series standard results
e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
\ln(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots \quad \text{for } -1 < x
\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots
\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots
(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots
\quad \text{for } |x| < 1
| Question | Answer |
|---|
| What is the Maclaurin series for e^x? | e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots |
| What is the Maclaurin series for \ln(1+x)? | \ln(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots |
| For what range of x does the Maclaurin series for \ln(1+x) converge? | It converges for -1 < x (i.e. x > -1). |
| What is the Maclaurin series for \sin x? | \sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots |
| What is the Maclaurin series for \cos x? | \cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots |
| What is the general term of the Maclaurin series for (1+x)^n? | (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots |
| For what values of x does the Maclaurin series for (1+x)^n converge? | It converges for $ |