# Section 6.4 Calculus 2

Taylor's Formula

## 6.4 Taylor’s Formula

### 6.4.1 Taylor’s Formula

• Taylor’s Formula guarantees that if $$f$$ has derivatives of all orders on an open interval containing $$a$$, then for every nonnegative integer $$n$$ and $$x$$ in that same interval, $$f(x)=\left(\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\right)+R_n(x)$$ where the error term is given by $$R_n(x)=\frac{f^{(n+1)}(x_n)}{(n+1)!}(x-a)^{n+1}$$ for some number $$x_n$$ between $$a$$ and $$x$$.
• Example Use the fact that $$e<4$$ and Taylor’s Formula to estimate the value of $$\sqrt{e}$$ with an error no greater than $$0.01$$.
• Example Use Taylor’s Formula to estimate the value of $$\sin(0.1)$$ with an error no greater than $$0.0001$$.

### 6.4.2 Convergence of Taylor and Maclaurin Series

• A Taylor series converges to its generating function when $$\lim_{n\to\infty}|R_n(x)|=0$$.
• Example Prove that $$e^x=\sum_{k=0}^\infty\frac{x^k}{k!}$$.
• Example Prove that $$\cos(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}$$.

### Review Exercises

1. Use the fact that $$e<3$$ and Taylor’s Formula to estimate the value of $$e$$ with an error no greater than $$0.001$$.
2. Use Taylor’s Formula to estimate the value of $$\cos(0.1)$$ with an error no greater than $$0.0001$$.
3. Use Taylor’s Formula to estimate the value of $$\sin(1)$$ with an error no greater than $$0.01$$.
4. Prove that $$\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$.
5. Use the fact that $$|\sinh(x_n)|\leq|\cosh(x_n)|\leq\cosh(x)$$ for any $$x_n$$ between $$0$$ and $$x$$ to prove that $$\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$$.
6. Reprove $$\cosh(x)=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$$ by using its definition $$\cosh(x)=\frac{1}{2}(e^x+e^{-x})$$ along with the Maclaurin series for $$e^x$$.

Solutions