Give an example of an infinite series of functions that converges to a limit fun

Question

Give an example of an infinite series of functions that converges to a limit function, and for which the derivative of the sum, term by term is not equal to the derivative of the limit function. (Hint: consider power series representations of functions and regions of convergence.)

Explanation / Answer

Examples A geometric series is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). Example: In general, the geometric series converges if and only if |z| < 1. An Arithmetico-geometric sequence is a generalization of the geometric series, which has coefficients of the common ratio equal to the terms in an arithmetic series. Example: The harmonic series is the series The harmonic series is divergent. An alternating series is a series where terms alternate signs. Example: The p-series converges if r > 1 and diverges for r = 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function. A telescoping series converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 - L.

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