Fourier Series of Piecewise Function | Odd Squares Sum π²/8 | Lecture 12

Learn Fourier Series of a piecewise-defined function on (−π, π), explained step by step in Hindi. In Lecture 12, Prof. Bharat Makhija solves Q15 in detail and uses the Fourier expansion to prove: 1/1² + 1/3² + 1/5² + ⋯ = π²/8 FUNCTION GIVEN IN Q15 f(x) = −π on the interval (−π, 0) f(x) = x on the interval (0, π) TOPICS COVERED • Type 1 Fourier Series — 2π interval • Q15: Fourier expansion of a piecewise-defined function • Setting up separate integrals on (−π, 0) and (0, π) • Calculating Fourier coefficients a0, an and bn • Integrating the constant and linear expressions • Simplifying coefficients using trigonometric values • Writing the complete Fourier expansion • Choosing a suitable substitution to deduce the series result • Proving that the sum of reciprocals of odd squares equals π²/8 Understand how to calculate Fourier coefficients when a function has different expressions on different intervals. Follow the detailed integration and simplification, then see how the resulting expansion gives an important infinite-series identity. ABOUT THE COMPLETE COURSE Learn Fourier Series from basics to advanced through 25 whiteboard lectures in Hindi, featuring 40+ solved problems across the complete playlist: 1. Type 1: Fourier Series with period 2π 2. Type 2: Fourier Series with period 2L — change of interval 3. Type 3: Half Range Fourier Series — sine and cosine series Useful for Engineering Mathematics, B.Sc. Mathematics and Mathematical Physics students, and for building foundations for relevant GATE papers. REFERENCE TEXTBOOKS • B. S. Grewal • N. P. Bali • B. V. Ramana ABOUT THE INSTRUCTOR Bharat Makhija Assistant Professor | B.Tech., M.E., Ph.D. (Pursuing) 12 years of teaching experience COMPLETE FOURIER SERIES PLAYLIST    • Fourier Series in Hindi | B.S. Grewal & N....   GOOGLE DRIVE RESOURCES https://drive.google.com/drive/folder... FOLLOW ON INSTAGRAM https://www.instagram.com/bharat_makh... Share your doubts about Q15, piecewise integration, or the series deduction in the comments. Subscribe for upcoming lectures. #FourierSeries #FourierSeriesInHindi #InfiniteSeries