Taylor & Maclaurin Series
\[
\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n
\qquad \qquad \qquad
\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
\]
-
Calculate the MacLaurin series
for the exponential function \(f(x) = \mathrm{e}^x.\)
Then notice that by evaluating this series at \(x=1,\)
we have a novel series for the constant \(\mathrm{e}.\)
-
We already knew a series for \(f(x) = \frac{1}{1-x}\)
because we recognized it as a geometric series.
However we should verify that this series
\[1+ x+x^2+x^3+x^4+x^5+\dotsb\]
is, in fact, the Maclaurin series for \(f(x) = \frac{1}{1-x}\)
manually, using the formula.
-
By altering the Taylor series
\[\frac{1}{1-x} = 1+ x+x^2+x^3+x^4+x^5+\dotsb\,,\]
find power series representations for the following functions:
\[
a(x) = \frac{1}{1-4x^2}
\qquad \qquad
b(x) = \frac{2}{3-x}
\qquad \qquad
c(x) = \frac{1}{1+x}
\]
\[
f(x) = \ln|1+x|
\qquad \qquad \qquad
g(x) = \frac{x}{(1-9x)^2}
\]
\[ h(x) = x^2\arctan\left(x^3\right) \]
What are the radii of convergence of these functions?
-
Find the MacLaurin series for sine and cosine,
and their respective radii of convergence.