Editing Chebyshev polynomials (section)

===Partial sums===
The partial sums of:
<math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x)</math>
are very useful in the [[approximation theory|approximation]] of various functions and in the solution of [[differential equation]]s (see [[spectral method]]). Two common methods for determining the coefficients {{mvar|a<sub>n</sub>}} are through the use of the [[inner product]] as in [[Galerkin's method]] and through the use of [[collocation method|collocation]] which is related to [[interpolation]].

As an interpolant, the {{mvar|N}} coefficients of the {{math|(''N''&nbsp;−&thinsp;1)}}st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto<ref>{{Cite web |url=http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |title=Chebyshev Interpolation: An Interactive Tour |access-date=2016-06-02 |archive-url=https://web.archive.org/web/20170318214311/http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |archive-date=2017-03-18 |url-status=dead }}</ref> points (or Lobatto grid), which results in minimum error and avoids [[Runge's phenomenon]] associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
<math display="block">x_k = -\cos\left(\frac{k \pi}{N - 1}\right); \qquad k = 0, 1, \dots, N - 1.</math>