Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz2 => Topic started by: Victor Ivrii on October 05, 2018, 06:15:41 PM

Show that each of the following series converges for all $z$:
\begin{align*}
&\sum_{n=0}^\infty \frac{z^n}{n!}, && \sum_{n=0}^\infty (1)^n \frac{z^{2n}}{(2n)!}.
\end{align*}

(1)
Apply ratio test:
[Z^n+1 / (n+1)!] / [Z^n / n!]=Z / n+1
Limit Z / n+1= 0 < 1 as n approaches infinity
thus it converges for all z
(2)
Apply ratio test:
[(1)^n+1*Z^2(n+1) / (2n+1)!]/ [(1)^n*Z^2n / (2n)!] = Z^2 / 2n+1
Limit Z^2 / 2n+1 = 0 < 1 as n approaches infinity.
thus it converges for all z
Beyond readability (and sanity)

Every power series has a convergence radius R, where Sum[a_{n}x^{n}] converges if x < R.
The first summation is the power series equal to e^{z}
and we have a_{n} = 1/n!
lim a_{n+1} / a^{n} = 1 / (n + 1)
= 0. So our convergence radius R is infinity and the power series converges for all z.
The second summation is the power series equal to cos(z) and we know:
cos(z) = (1/2)*(e^{iz} + e^{iz})
From the first series, we know the e^{z} is convergent on all z, so cos(z) is also convergent on all z.

In attachment.

Jeff, learn a bit of LaTeX, since without it anything but the most simple math expressions will be out of your reach
http://forum.math.toronto.edu/index.php?topic=610.0 (http://forum.math.toronto.edu/index.php?topic=610.0)
Alex, learn how to scan properly