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lants
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Homework Statement
Determine for what k f(x)=xk is an element of L2 (0,1) vector space
k ∈ ℝ
Homework Equations
The Attempt at a Solution
[tex] \int_{0}^{1} x^{2k} dx = \frac{1-0^{2k+1}}{1+2k} = \sum_{n=0}^{\infty}{(-2k)^{n}} [/tex] (for k > -½)
This sum should converge for [tex]
\lim_{n \to +\infty}
{\frac{|(-2k)^{n+1}|}{|(-2k)^{n}|}} < 1
=
|-2k| < 1
[/tex]
Which gives me a radius of convergence for
[tex] - \frac{1}{2} < k < \frac{1}{2}
[/tex]
But just by examining it, the integral should exist for any k greater than negative one-half, what is wrong with my ratio test?
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