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Homework5 - ∑ ∞ 1(2 n-1-6 Hint Use the Fourier...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 6 – 13 November 2007 1. Parseval’s equality says that if f is Riemann integrable on ( - π, π ) and has Fourier series ( a 0 / 2) + n 1 ( a n cos nx + b n sin nx ) = + n = -∞ c n e inx , then 1 2 π π - π | f ( x ) | 2 dx = 1 4 | a 0 | 2 + 1 2 n =1 ( | a n | 2 + | b n | 2 ) = n = -∞ | c n | 2 . Using this fact, find the value of the series 1 n 2 ( n 2 +1)
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Unformatted text preview: ∑ ∞ 1 (2 n-1)-6 . Hint: Use the Fourier expansions of the functions f ( x ) = sinh x , respectively g ( x ) = x ( π- | x | ), (-π < x ≤ π ). 2. Let l > 0. Show that    s 2 l sin ± n-1 2 ² πx l    ∞ 1 is an orthonormal set on PC (0 , l )....
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