Let \( f_j : A \to \mathbb{R}, j = 1, 2, \ldots \) be a sequence of functions. When we say that the series \( \sum_{j=1}^{\infty} f_j \) converges uniformly to \( f \), we mean that: - \( s_n \to f \) uniformly, where \( s_n = \sum_{j=1}^{n} f_j \), the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j \). - \( s_n(x) \to f(x) \) for each \( x \in A \), where \( s_n(x) = \sum_{j=1}^{n} f_j(x), x \in A \) is the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j(x) \). - \( f_n \to f \) uniformly. - \( f_j(x) \to f(x) \) as \( j \to \infty \) for each \( x \in A \).
Let \( f_j : A \to \mathbb{R}, j = 1, 2, \ldots \) be a sequence of functions. When we say that the series \( \sum_{j=1}^{\infty} f_j \) converges uniformly to \( f \), we mean that: - \( s_n \to f \) uniformly, where \( s_n = \sum_{j=1}^{n} f_j \), the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j \). - \( s_n(x) \to f(x) \) for each \( x \in A \), where \( s_n(x) = \sum_{j=1}^{n} f_j(x), x \in A \) is the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j(x) \). - \( f_n \to f \) uniformly. - \( f_j(x) \to f(x) \) as \( j \to \infty \) for each \( x \in A \).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let \( f_j : A \to \mathbb{R}, j = 1, 2, \ldots \) be a sequence of functions. When we say that the series \( \sum_{j=1}^{\infty} f_j \) converges uniformly to \( f \), we mean that:
- \( s_n \to f \) uniformly, where \( s_n = \sum_{j=1}^{n} f_j \), the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j \).
- \( s_n(x) \to f(x) \) for each \( x \in A \), where \( s_n(x) = \sum_{j=1}^{n} f_j(x), x \in A \) is the nth partial sum of the series \( \sum_{j=1}^{\infty} f_j(x) \).
- \( f_n \to f \) uniformly.
- \( f_j(x) \to f(x) \) as \( j \to \infty \) for each \( x \in A \).
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