Differentiability of a Series of Functions

In summary, the problem at hand is to show that the series of functions, f(x) = Ʃ (xn)/n2, where n≥1, converges to some f(x) and that f(x) is continuous, differentiable and integrable on the interval [-1,1]. It is known that f(x) is continuous and integrable, but the challenge lies in proving its differentiability. The series of derivatives converges uniformly on the interval (-1,1), which suggests that the original series is differentiable there. However, it may not be differentiable at the endpoints, 1 and -1. The Weierstrass M-Test may not be applicable in this case.
  • #1
luke8ball
22
0
I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (xn)/n2, where n≥1, converges to some f(x), and that f(x) is continuous, differentiable, and integrable on [-1,1].

I know how to show that f(x) is continuous, since each fn(x) is continuous, and I fn(x) converges uniformly. Because each fn(x) is also integrable, I can also show f(x) is integrable.

The trouble I'm having is proving that f(x) is differentiable. I need to show that the series of derivatives converges uniformly. However, I don't think I can use the Weierstrass M-Test in this scenario. Any ideas?
 
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  • #2
luke8ball said:
I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (xn)/n2, where n≥1, converges to some f(x), and that f(x) is continuous, differentiable, and integrable on [-1,1].

I know how to show that f(x) is continuous, since each fn(x) is continuous, and I fn(x) converges uniformly. Because each fn(x) is also integrable, I can also show f(x) is integrable.

The trouble I'm having is proving that f(x) is differentiable. I need to show that the series of derivatives converges uniformly. However, I don't think I can use the Weierstrass M-Test in this scenario. Any ideas?
The differentiated series converges on the open interval (-1,1). Hence the original series is differentiable there and the derivative is the differentiated series. The differentated series diverges at 1 and converges conditionally at -1. Although I cannot prove it, this makes me suspect that the original series is not differentiable at 1 and -1.
 

1. What is the definition of differentiability for a series of functions?

Differentiability of a series of functions refers to the ability of each individual function in the series to have a derivative at a given point. This means that the function is smooth and continuous at that point, and the slope of the tangent line can be calculated.

2. How is differentiability related to continuity?

A function must be continuous in order to be differentiable. This means that the function must be defined and have a limit at a given point. If a function is not continuous, it cannot have a derivative and therefore is not differentiable.

3. What is the difference between pointwise differentiability and uniform differentiability in a series of functions?

Pointwise differentiability refers to the differentiability of each individual function in a series at a given point. Uniform differentiability, on the other hand, refers to the differentiability of the entire series of functions as a whole. A series of functions is uniformly differentiable if the derivatives of each function in the series converge uniformly.

4. Can a series of non-differentiable functions be differentiable?

Yes, a series of non-differentiable functions can still be differentiable. This can occur if the functions are not continuous at a given point, but the series as a whole is still continuous and differentiable. An example of this is the Weierstrass function, which is continuous but not differentiable at any point.

5. What are some applications of differentiability of a series of functions in real-life situations?

Differentiability of a series of functions has many practical applications, including in the fields of physics, engineering, and economics. It is used to calculate rates of change, determine the behavior of systems over time, and optimize processes. For example, in economics, the concept of differentiability is used to calculate marginal cost and marginal revenue, which are important in making business decisions.

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