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We have developed precise analytic definitions of the convergence of a sequence and continuity of a function and we have used these to prove the EVT and IVT for a continuous function. Moreover, if we differentiate or integrate these series term by term then the resulting series will converge to the derivative or integral of the original series. This was not always the case for Fourier series. For example consider the function.

The partial sums for a Taylor series were polynomials and hence continuous but what they converged to was continuous as well. The difficulty is quite delicate and it took mathematicians a while to determine the problem. There are two very subtly different ways that a sequence of functions can converge: pointwise or uniformly. This distinction was touched upon by Niels Henrik Abel in while studying the domain of convergence of a power series.

However, the necessary formal definitions were not made explicit until Weierstrass did it in his paper Zur Theorie der Potenzreihen On the Theory of Power Series. This was published in his collected works in It will be instructive to take a look at an argument that doesn't quite work before looking at the formal definitions we will need. Of course, it is obvious to us that this is not true because we've seen several counterexamples.

But Cauchy, who was a first rate mathematician was so sure of the correctness of his argument that he included it in his textbook on analysis, Cours d'analyse This is the type of convergence we have been observing to this point.

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By contrast we have the following new definition. The difference between these two definitions is subtle.

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However the reverse is not true. This will become evident, but first consider the following example. This never happens with a power series, since they converge to continuous functions whenever they converge.

We will also see that uniform convergence is what allows us to integrate and differentiate a power series term by term. Section In the mathematical field of analysisuniform convergence is a mode of convergence of functions stronger than pointwise convergence. The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning.

In Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in found purported counterexamples in the context of Fourier seriesarguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit.

The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis.

We first define uniform convergence for real-valued functionsalthough the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces see below. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.

In this situation, uniform limit of continuous functions remains continuous. Uniform convergence admits a simplified definition in a hyperreal setting. Given a topological space Xwe can equip the space of bounded real or complex -valued functions over X with the uniform norm topology, with the uniform metric defined by. Then uniform convergence simply means convergence in the uniform norm topology:.

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These observations preclude the possibility of uniform convergence. In fact, it is easy to see that. In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. Theorem Weierstrass M-test. The following result states that continuity is preserved by uniform convergence:. This theorem is an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a Fourier series of continuous functions.

The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous originally stated in terms of convergent series of continuous functions is infamously known as "Cauchy's wrong theorem".

The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable not even if the sequence consists of everywhere- analytic functions, see Weierstrass functionand even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives.Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

I fail to see how this is true. Suppose that the latter case holds. Hence, the power series contains only finitely many terms. I'll use this in two places. Sign up to join this community. The best answers are voted up and rise to the top. Asked 4 years, 6 months ago. Active 4 years, 6 months ago. Viewed 2k times. Improve this question. Yoni Yoni 3 3 silver badges 15 15 bronze badges. What precisely does it mean to be uniformly convergent on the reals?

Related 4.The definition of a uniformly-convergent series is equivalent to the condition. A sufficient condition for the uniform convergence of a series is given by the Weierstrass criterion for uniform convergence.

There are criteria for the uniform convergence of series analogous to Dirichlet's and Abel's criteria for the convergence of series of numbers. These tests for uniform convergence first occurred in papers of G. If in a series. Continuity of the sum of a series.

### 8.1: Uniform Convergence

In the study of the sum of a series of functions, the notion of "point of uniform convergence" turns out to be useful. Therefore, if a series of continuous functions converges uniformly on a topological space, then its sum is continuous on that space. Hence it follows that the sum of any series of continuous functions, convergent in some interval, is continuous on a dense set of points of the interval.

At the same time there exists a series of continuous functions, convergent at all points of an interval, such that the points at which it converges non-uniformly form an everywhere-dense set in the interval in question.

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Term-by-term integration of uniformly-convergent series. If the terms of the series. At the same time there are various generalizations.

## Uniformly-convergent series

Below some results for the Stieltjes integral are given. Conditions for term-by-term differentiation of series in terms of uniform convergence. In this way, the presence of the property of uniform convergence of a series, in much the same way as absolute convergence see Absolutely convergent seriespermits one to transfer to these series certain rules of operating with finite sums: for uniform convergence — term-by-term passage to the limit, term-by-term integration and differentiation see 3 — 6and for absolute convergence — the possibility of permuting the order of the terms of the series without changing the sum, and multiplying series term-by-term.

The properties of absolute and uniform convergence for series of functions are independent of each other. Thus, the series. Log in. Namespaces Page Discussion.

### Uniform convergence

Views View View source History. Jump to: navigationsearch. Properties of uniformly-convergent series. Formula 5 has been generalized to functions of several variables. For references see Series. How to Cite This Entry: Uniformly-convergent series.

Encyclopedia of Mathematics. This article was adapted from an original article by L.Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Hint: Any partial sum is bounded, but the series converges pointwise to something unbounded. Do you see a contradiction? Sign up to join this community. The best answers are voted up and rise to the top. Asked 3 days ago. Active 3 days ago.

Arthur Arthur k 13 13 gold badges silver badges bronze badges. Show that a sequence of bounded functions can't converge uniformly to an unbounded function. So you don't need to specify "the uniform limit" or "the pointwise limit". There is just "the limit". And in this case we can show, without too much effort, that the limit is unbounded. With a little more effort, we can show that if it converges uniformly, then the limit is bounded. From these two facts we can easily conclude that the sequence can't converge uniformly.

There is no counter-example. But this ought to work too.

Three days grace songs in drop d Find each of the following infinite sums. Evaluate the following infinite sums. To evaluate the various power series, manipulate them until some well-known power series emerge. However, a beautiful theorem of Abel shows that the series does converge uniformly on [0,1].

Find a sequence which is Abel summable, but which is not summable. Hint: Look over the list of Taylor series until you find one which does not converge at 1 even though the function it represents is continuous at 1.

Since we already know examples of discontinuous derivatives, this provides another example where the pointwise limit of continuous functions is not continuous. This problem outlines a completely different approach to the integral; consequently, it is unfair to use any facts about integrals learned previously. The ouly remaining question is: Which functions are regulated? Conclude that every regulated function has the same property, and find an integrable function that is not regulated.

Calculus 4th Michael Spivak. Chapter 23 Uniform Convergence and power Series. Chapter Questions. Problem 3 Find the Taylor series at 0 for each of the following functions.

Problem 4 Find each of the following infinite sums. Problem 5 Evaluate the following infinite sums. Problem 21 a Using Problem 19find the following infinite sums. Problem 30 This problem outlines a completely different approach to the integral; consequently, it is unfair to use any facts about integrals learned previously.Download the video from iTunes U or the Internet Archive.

Topics covered: Weirstrass M-test; using power series to evaluate definite integrals when we do not know the anti-derivative of the integrand. This section contains documents that are inaccessible to screen reader software. A " " symbol is used to denote such documents.

Today is the day we've all been waiting for. We've come to the last lecture of our course, and it's, I think, a rather satisfying lecture today, not just because it is the last one, but content-wise too.

Today we're going to talk about uniform convergence of series. Remember, a series can be viewed as a sequence of partial sums, and consequently our discussion on uniform convergence applies here. And what we're going to do-- you see, again, it's the same old story. Now that we know what the concept means, is there an easy way to tell when we have the property? And I figure again that, this being the last lecture, I should give you some big names to remember.

And name-dropping-wise, I come to our first concept today, which I call-- I don't call it that. It's named the Weierstrauss M-test.

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The Weierstrauss M-test is a very, very convenient method for determining whether a given series converges uniformly or not. By the way, let me just make one little aside. Instead of saying the sequence of partial sums converges uniformly to the infinite series, we usually abbreviate that simply by saying, the infinite series is uniformly convergent. Let me just say that one more time. If I say that the series is uniformly convergent, that's an abbreviation for saying that the sequence of partial sums converges uniformly to the series.

But at any rate, let me now go over the so-called Weierstrauss M-test with you. It's a rather simple test. I will give you both the proof of the test and some applications of it. The test simply says this-- and this is where the name M-test comes from.