What are the limits of the convergence of the Maclaurin series for $e^x$ regarding error bounds?

Conditions

• The function is $f(x) = e^x$.
• The series is the Maclaurin expansion centered at 0.
• Considering pointwise and uniform convergence properties.

Reasoning, step by step

1. Acknowledge that the series converges pointwise for all $x \in \mathbb{R}$.
2. Acknowledge uniform convergence on any bounded interval $[-M, M]$.
3. Note that 'uniform convergence on bounded intervals' is not the same as 'uniform convergence on $\mathbb{R}$'.
4. Recognize that adding a term improves the match of derivatives at $x=0$, but does not strictly reduce error everywhere simultaneously.
5. Conclude that global error bounds are not uniform across the entire real line.

Example

The script states: 'For this function the series converges at every fixed real x, uniformly on each fixed bounded interval. This does not promise a uniformly small error over the entire real line or a strictly smaller error at every point after every added term.'

Common misconceptions

• Believing that pointwise convergence implies uniform convergence on the whole real line.
• Assuming that each new term strictly reduces the error at every $x$ compared to the previous partial sum.

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Connected concepts

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