Taylor series

Consider the infinite Taylor series given by

1. As described in the statement of the Ratio Test, let $r_n(x)$ be the ratio of the $(n+1)^{\text{st}}$ term of $T(x)$ to the $n^{\text{th}}$ term of $T(x)$. Find the simplest formula that you can for $r_n(x)$.

2. Let $r(x) = \lim_{n \to \infty} r_n(x)$. Evaluate this limit to find the simplest formula you can for $r(x)$.

3. For what values of $x$ is $|r(x)| \lt 1$? What is the open interval of convergence for $T(x)$?

4. Let $T_{10}(x)$ be the sum of the first $10$ terms of $T(x)$, and let $f(x) = \ln(2) - \ln(3-x)$. Plot…

Let $f(x) = e^x$. Our goal is to understand why the Taylor series for $f(x)$ converges for every real number $x$ and see that the Taylor series converges to $e^x$.

1. For $f(x) = e^x$, explain why $f^{(k)}(0) = 1$ for every natural number $k$.

2. State the Taylor series, $T_f(x)$ centered at $a = 0$ for $f(x) = e^x$. Write $T_f(x)$ in both sigma notation and as an expanded sum.

3. Let $r_n(x)$ be the ratio of the $(n+1)^{\text{st}}$ term to the $n^{\text{th}}$ term of $T_f(x)$. Find the simplest expression you can for $r_n(x)$.

4. Let $r(x) = \lim_{n \to \infty} r_n(x)$. Evalua…

The examples we have considered so far in this section have all been for Taylor polynomials and series centered at 0, but Taylor polynomials and series can be centered at any value of $a$.

1. Let $f(x) = \cos(x)$. Find the Taylor polynomials up through order four of $f$ centered at $a = \frac{\pi}{2}$. Then find the Taylor series for $f(x)$ centered at $a = \frac{\pi}{2}$. Why is the result not surprising?

2. Let $f(x) = \frac{1}{1+x} = (1+x)^{-1}$. Find the Taylor polynomials up through order fou…

As we will see in more detail in the next section, we can use known Taylor series to obtain other Taylor series, and we preview that idea in this exercise.

1. Calculate the first four derivatives of $\sin(x^2)$ and hence find the fourth order Taylor polynomial for $\sin(x^2)$ centered at $a=0$.

2. Part (a) demonstrates the direct approach to finding Taylor polynomials and series. Next we utilize a known Taylor series to make the…