Problem 1: (20 points) Write the McLaurin series of the functions listed below.
Indicate the interval of convergence.
• $f(x) = e^{x^2}$
• $f(x) = \frac{1}{1-x^2}$
• $f(x) = x^2 \sin(x)$
Problem 2: (15 points) Find the Taylor series of $f(x) = \frac{1}{1-x}$ with center at
$c = 3$. Indicate the interval of convergence.
Problem 3: (10 points) Write the first four terms of the Taylor series of $y = f(x)$
with center $c = -1$ if $f(-1) = 0$, $f'(-1) = 3$, $f''(-1) = 1$, $f^{(3)}(-1) = \frac{1}{2}$ and
$f^{(4)}(-1) = 2$.
Problem 4: (5 points) Indicate if the following statement is True or False.
• It is possible to find a power series $\sum_{0}^{\infty} b_n(x+1)^n$ that converge on the interval
$(-2,2)$ and diverge outside that interval. If not, explain why.
Problem 5: (15 points) Find the radius of convergence of $\sum_{0}^{\infty} \frac{(x-2)^n}{(2n+1)5^n}$. DO
NOT study the convergence at the end-points of the interval.
Problem 6: (15 points) Use the power series of $(1+x^4)^{-1}$ and differentiation to
find the Power series expansion of $f(x) = \frac{4x^3}{(x^4+1)^2}$ for $|x| < 1$.
Problem 7: (15 points)
Compute the McLaurin series of $f(x) = \int \frac{\sin(x)}{x} dx$
