Summary

1. Infinite series - convergence tests, series with alternating signs, absolute convergence, Dirichlet and Abel tests, Riemann's theorem, indefinite integrals. 2. Sequences and series of functions, pointwise and uniform convergence, Weierstrass' test, power series, Taylor series. 3. Fourier series - the space of piecewise continuous functions with inner product, norm convergence, orthonormal and orthogonal bases, Fourier series, sine and cosine series, convergence of Fourier series. 4. Orthogonal polynomials - Legendre, Hermite, Laguerre 5. The Fourier transform - definition, conditions of convergence, the inversion formula, basic properties of the transform, convolution, differentiation under the integral sign.