Toronto Math Forum
MAT334-2018F => MAT334--Tests => Quiz-2 => Topic started by: Victor Ivrii on October 05, 2018, 06:15:41 PM
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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*}
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(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)
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Every power series has a convergence radius R, where Sum[anxn] converges if |x| < R.
The first summation is the power series equal to ez
and we have an = 1/n!
lim |an+1| / |an| = 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)*(eiz + e-iz)
From the first series, we know the ez is convergent on all z, so cos(z) is also convergent on all z.
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In attachment.
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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