Function represented by power series

In summary, to prove that a function f represented by a power series with a radius of convergence )<R<∞ is continuous on the interval (a-R, a+R), it is necessary to use the fact that a power series is uniformly continuous within its radius of convergence. Simply showing that the series converges on the interval does not prove continuity. A counterexample can be constructed using a series of continuous functions that converges to a non-continuous function.
  • #1
sandra1
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Homework Statement


If a function f is represented by the power series ∑(k=0 to ∞) a_k(x-a)^k, with a radius of convergence )<R<∞, then f is continuous on the interval (a-R, a+R)


Homework Equations





The Attempt at a Solution



I don't know if my proof is loose at some point or not, because it seems so easy. Is there anything that is obviously not right about my proof? Thanks very much

So choose x=m in (a-R, a+R), the series converges on the interval (a-R,a+R)
then we would have
lim(x->m) ∑(k=0 to ∞) a_k(x-a)^k = ∑(k=0 to ∞) (a_k)lim(x->m) (x-a)^k = ∑(k=0 to ∞) (a_k)(m-a)^k = f(m).

So f is continuous on any arbitrary m on (a-R, a+R) therefore ∑(k=0 to ∞) a_k(x-a)^k is continuous on (a-R, a+R)
 
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  • #2
It is possible to have series of continuous functions that converges to a non-continuous function. To prove the result, you must use the fact that a power series is uniformly continuous within its radius of convergence.
 
  • #3
Your proof is invalid, because you cannot assume that

[tex]\lim_{x \to m} \lim_{n \to \infty} \sum_{k=0}^n a_k (x-a)^k = \lim_{n \to \infty} \lim_{x \to m} \sum_{k=0}^n a_k (x-a)^k[/tex].

Consider [itex]x_{m,n}[/itex] = 0 when n < m and 1 when n >= m. Then [itex]\lim_{n \to \infty} \lim_{m \to \infty} x_{m,n} \neq \lim_{m \to \infty} \lim_{n \to \infty} x_{m,n}[/itex].
 
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1. What is a function represented by a power series?

A function represented by a power series is a mathematical expression that can be written as an infinite sum of powers of a variable, typically x. It is used to approximate other functions and can be used to find the value of a function at a specific point.

2. How is a power series different from a regular polynomial?

A power series has an infinite number of terms, while a regular polynomial has a finite number of terms. Additionally, the powers in a power series increase by one with each term, while the powers in a polynomial can vary.

3. What is the purpose of using a power series to approximate a function?

The purpose of using a power series to approximate a function is to simplify the calculation of the function at a specific point. By using a power series, we can avoid complex calculations and still get a good estimate of the function's value at the desired point.

4. Can a power series represent any function?

No, a power series can only represent functions that are analytic, meaning they can be expressed as power series themselves. This includes most common functions such as polynomials, trigonometric functions, and exponential functions.

5. How do you determine the convergence of a power series?

The convergence of a power series can be determined by finding the radius of convergence, which is the distance from the center of the series to the nearest point where the series does not converge. This can be done using various convergence tests, such as the ratio test or the root test.

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