Problems

01

Approximating $1/e$

Using the series representation of $e^x$, show that:

$$\frac{1}{e}=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots$$

Now use the alternating series error bound to approximate $\frac{1}{e}$ to an error within ${10}^{-3}$.

02

Some estimates using series

Without a calculator, estimate $\cos (0.02)$ (angle in radians) with an error below ${10}^{-6}$.

(Use the error bound formula for alternating series.)

03

Some estimates using series

Find an infinite series representation of ${\displaystyle {\int }_0^1\sin (x^2)dx}$ and then use your series to estimate this integral to within an error of ${10}^{-3}$.

(Use the error bound formula for alternating series.)