Answer
(a) The expression for the n-th degree Taylor polynomial of $f$ centered at $a$ is: $T_n=\Sigma_{i=0}^n\frac{f^(i)(a)}{i!}(x-a)^i$.
(b) The expression for Taylor series of $f$ centered at $a$ is: $\Sigma_{n=0}^n\frac{f^(n)(a)}{n!}(x-a)^n$.
(c) The expression for Maclaurin series of $f$ is: $\Sigma_{n=0}^n\frac{f^{(n)}(0)}{n!}(x)^n$.
(d) The expression for the n-th degree Taylor polynomial of $f$ centered at $a$ is: $T_n=\Sigma_{i=0}^n\frac{f^(i)(a)}{i!}(x-a)^i$, and that $f(x)=T_n(x)+R_n(x).
If \lim\limits_{n \to \infty} R_n(x)=0$ for $|x-a| \lt R$ , then $f$ is equal to the sum of its Taylor's series.
(e) Taylor's inequality states that if $|f^{n+1}(x) \leq M|$ for $|x-a| \leq d$, then the remainder $R_n(x)$ of the Taylor series satisfies the inequality described as follows:$|R_n(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}$ for $|x-a|\leq d$.
Work Step by Step
(a) The expression for the n-th degree Taylor polynomial of $f$ centered at $a$ is: $T_n=\Sigma_{i=0}^n\frac{f^(i)(a)}{i!}(x-a)^i$.
(b) The expression for Taylor series of $f$ centered at $a$ is: $\Sigma_{n=0}^n\frac{f^(n)(a)}{n!}(x-a)^n$.
(c) The expression for Maclaurin series of $f$ is: $\Sigma_{n=0}^n\frac{f^{(n)}(0)}{n!}(x)^n$.
(d) The expression for the n-th degree Taylor polynomial of $f$ centered at $a$ is: $T_n=\Sigma_{i=0}^n\frac{f^(i)(a)}{i!}(x-a)^i$, and that $f(x)=T_n(x)+R_n(x).
If \lim\limits_{n \to \infty} R_n(x)=0$ for $|x-a| \lt R$ , then $f$ is equal to the sum of its Taylor's series.
(e) Taylor's inequality states that if $|f^{n+1}(x) \leq M|$ for $|x-a| \leq d$, then the remainder $R_n(x)$ of the Taylor series satisfies the inequality described as follows:$|R_n(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}$ for $|x-a|\leq d$.
