Convergence of Fourier Series for f(t) = 1 + t with Only Cosine Terms in [0,pi]

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SUMMARY

The Fourier series expansion of the function f(t) = 1 + t, using only cosine terms in the interval [0, π], converges to the function except at the discontinuities. The series converges to the value of 1 + π/2 at the endpoints t = 0 and t = π, which differs from the original function f(t) at those points. The convergence occurs for t in the open interval (0, π).

PREREQUISITES
  • Understanding of Fourier series and their properties
  • Knowledge of even functions and their expansions
  • Familiarity with convergence concepts in mathematical analysis
  • Basic calculus, particularly limits and continuity
NEXT STEPS
  • Study the properties of Fourier series convergence, specifically for piecewise continuous functions
  • Explore the concept of even and odd functions in Fourier analysis
  • Learn about the Dirichlet conditions for Fourier series convergence
  • Investigate the implications of convergence at points of discontinuity
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Mathematics students, educators, and anyone interested in advanced calculus or Fourier analysis, particularly those focusing on series convergence and function representation.

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Homework Statement



Write f(t) = 1 + t as Fourier series, with only cosine terms in the interval [0,pi]

For which values of t does the series converge to f ?



The Attempt at a Solution



Expand f = 1+t as an even function about t=0; so it will be a zig-zag with non continuous points at -pi,0,pi, 2pi etc

the first part is simple, but the second; I was thinking that the Fourier series is converging to f, exept where f is not continous. So the answer is [tex]t \in ]0,\pi[[/tex]

Is that correct?
 
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Yes. In fact it converges to 1+pi/2 at 0 and pi which is not the same as f.
 

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