11. Whenever a series is Ces�ro summable, it is also Abel summable and has the same sum. 12. If a series is convergent, then it is Ces�ro summable and its Ces�ro sum is the usual sum. 13. Hence we say that any sequence a _ n is Ces�ro summable to some " a " if 14. The above definitions work for signals that are square integrable, or square summable , that is, of finite energy. 15. Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. 16. For any convergent sequence, the corresponding series is Ces�ro summable and the limit of the sequence coincides with the Ces�ro sum. 17. The implication is that the Fourier series of any continuous function is Ces�ro summable to the value of the function at every point. 18. Generally, the terms of a summable series should decrease to zero; even could be expressed as a limit of such series. 19. On the other hand, taking the Cauchy product of Grandi's series with itself yields a series which is Abel summable but not Ces�ro summable: 20. On the other hand, taking the Cauchy product of Grandi's series with itself yields a series which is Abel summable but not Ces�ro summable :
