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*open*disc?

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- Thread starter markiv
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[tex]\sum_{n = 0}^{\infty} x^n = \frac{1}{1 - x}[/tex]

for [itex]|x| < 1[/itex]. This is clearly unbounded as [itex]x[/itex] approaches 1.

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Mute

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[tex]\sum_{n = 0}^{\infty} x^n = \frac{1}{1 - x}[/tex]

for [itex]|x| < 1[/itex]. This is clearly unbounded as [itex]x[/itex] approaches 1.

Hm. If I recall correctly from my Complex variables class years ago, the geometric series actually is uniformly convergent on ##|z| < 1##. However, it is not uniformly convergent on ##|z| \leq 1##.

Do I remember incorrectly?

- #5

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The partial sums areHm. If I recall correctly from my Complex variables class years ago, the geometric series actually is uniformly convergent on ##|z| < 1##. However, it is not uniformly convergent on ##|z| \leq 1##.

Do I remember incorrectly?

[tex]s_N(x) = \sum_{n = 0}^{N-1}x^n = \frac{1 - x^{N}}{1 - x}[/tex]

The limit is

[tex]s(x) = \sum_{n = 0}^{\infty}x^n = \frac{1}{1 - x}[/tex]

So

[tex]s(x) - s_N(x) = \frac{x^N}{1 - x}[/tex]

This is the error in approximating the series by the N'th partial sum. For any fixed N, this error is arbitrarily large as [itex]x \rightarrow 1[/itex]. If the convergence were uniform, we would be able to uniformly bound the error as small as we like by making N large enough.

I think what you are remembering is that the convergence is uniform on [itex]|z| \leq R[/itex] for any positive [itex]R < 1[/itex].

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Thank you. This makes a lot of sense.

- #7

Mute

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The partial sums are

[tex]s_N(x) = \sum_{n = 0}^{N-1}x^n = \frac{1 - x^{N}}{1 - x}[/tex]

The limit is

[tex]s(x) = \sum_{n = 0}^{\infty}x^n = \frac{1}{1 - x}[/tex]

So

[tex]s(x) - s_N(x) = \frac{x^N}{1 - x}[/tex]

This is the error in approximating the series by the N'th partial sum. For any fixed N, this error is arbitrarily large as [itex]x \rightarrow 1[/itex]. If the convergence were uniform, we would be able to uniformly bound the error as small as we like by making N large enough.

I think what you are remembering is that the convergence is uniform on [itex]|z| \leq R[/itex] for any positive [itex]R < 1[/itex].

Yes, that's probably the result I was thinking of.

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