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ChemEng1
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Homework Statement
Determine whether the following sequence {xn} converges in ℂunder the usual norm.
[itex]x_{n}=n(e^{\frac{2i\pi}{n}}-1)[/itex]
Homework Equations
[itex]e^{i\pi}=cos(x)+isin(x)[/itex]
ε, [itex]\delta[/itex] Definition of convergence
The Attempt at a Solution
I would like some verification that this response answers the problem statement. I have not worked with complex numbers since high school algebra. So this attempt may be very wrong.
The professor stated that for complex number to converge, both the real and imaginary parts have to converge.
[itex]x_{n}=n(cos(\frac{2\pi}{n})+isin(\frac{2\pi}{n})-1)[/itex], by substitution with Euler's formula.
[itex]x_{n}=ncos(\frac{2\pi}{n})-n+isin(\frac{2\pi}{n})n[/itex]. This appears to converge to 0+0i as n→∞.
Real Part:[itex]Re(x_{n})=ncos(\frac{2\pi}{n})-n=>n(cos(\frac{2\pi}{n})-1)=>n*0=0[/itex]
Imaginary Part:[itex]Im(x_{n})=sin(\frac{2\pi}{n})n=>0*n=0[/itex]
Since both the real and imaginary parts of [itex]x_{n}[/itex] converge, then [itex]x_{n}[/itex] converges.
Is this close? Did I miss anything?
Thanks in advance,
Scott