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![]() We show how to find limits of sequences that converge, often by using the properties of limits for functions discussed earlier. We then say that \((f_n) = (f_1,f_2,f_3,\ldots,)\) is a sequence of functions on \(A\).Ĭonsider the sequence \((f_n)\) defined on \(\real\) by \(f_n(x) = (2xn+(-1)^nx^2)/n\). Glossary Contributors and Attributions In this section, we introduce sequences and define what it means for a sequence to converge or diverge. Let \(A\subset\real\) be a non-empty subset and suppose that for each \(n\in\N\) we have a function \(f_n:A\rightarrow\real\). In this section, we develop a notion of the limit of a sequence of functions and then investigate if the fundamental properties of boundedness, continuity, integrability, and differentiability are preserved under the limit operation. We will see that this latter issue is rather delicate. Consists of rational numbers (1, 3/2, 17/12,), which is clear from the definition however it converges to the irrational square root of 2, see Babylonian. Moreover, it would be desirable that the limiting function \(f\) inherit as many properties possessed by each function \(f_n\) such as, for example, continuity, differentiability, or integrability. In many of these types of problems, one is able to generate a sequence of functions \((f_n)=(f_1,f_2,f_3,\ldots)\) through some algorithmic process with the intention that the sequence of functions \((f_n)\) converges to the solution \(f\). A typical way that sequences of functions arise is in the problem of solving an equation in which the unknown is a function \(f\). ![]() Sequences of functions arise naturally in many applications in physics and engineering. ![]() In this section, we consider sequences whose terms are functions. In the previous sections, we have considered real-number sequences, that is, sequences \((x_n)\) such that \(x_n \in \real\) for each \(n\in\N\).
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