Summation by Parts: Lim x->1 (1-x)f(x)=L

In summary, we can use a "trick" to express |(1-x)f(x)-L| as a single power series. By breaking it into two summations and using the geometric series, we can show that this power series is continuous and has an upper bound, proving that |(1-x)f(x)-L| is small, as desired.
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tracedinair
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Homework Statement



Let lim n-> ∞ a_n = L. Then, let f(x) = ∑ from n=0 to ∞ of (a_n)(x^n). Show that the lim x-> 1 (1-x)f(x) = L.

Homework Equations


The Attempt at a Solution



This one is pretty far over my head. I know at some point you're supposed to use Abel/SBP, but here is what I have so far.

Let |a_n| go to |L|.

Then, using the ratio test, let |a_n|^(1/n) go to |L|^(1/∞) = |L|^(0) = 1.

Then, from the Ratio test, we can see that the series will converge for |x| < 1.

Take ∑ from 0 to ∞ of (a_n)(x^n). Then, multiply through. So, we obtain, (1-x)∑ from 0 to ∞ of (a_n)(x^n) = ∑ from 0 to ∞ of (x^n - x^(n+t)). Taking b_n to equal (x^n - x^(n+t)) we can get ∑ (a_n)(b_n)..

This is where I get stuck. I'm not really where to take it from here.
 
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  • #2
Here's my way. Start over. (You won't use Abel or SBP.)

You want to show |(1-x)f(x)-L| is small.

Use a "trick" that L=(1-x)L/(1-x) and write 1/(1-x) as the geometric series.

This enables you to express |(1-x)f(x)-L| as a single power series.

Since a_n approaches L, we have |a_n - L| < epsilon, or epsilon/2, or whatever you need, for n>N.

Break your power series for |(1-x)f(x)-L| into two summations, one up to N, the other for N+1 and beyond.

For the summation up to N, this is continuous at x=1.

For the summation starting at N+1, get an upper bound. You will again use the geometric series.

I hope I didn't give away too much or too little. This plan comes from looking at the proof of Abel's Theorem. In that proof, essentially one uses SBP to reduce the given problem to your problem.
 

Related to Summation by Parts: Lim x->1 (1-x)f(x)=L

What is Summation by Parts?

Summation by Parts is a method used in calculus to evaluate the limit of a function as it approaches a particular value, typically denoted by "x->1". It involves breaking down the function into smaller parts and using known limits and properties to simplify the overall expression.

What does the notation "Lim x->1" mean?

The notation "Lim x->1" is read as "the limit of x as it approaches 1". It represents the value that the function will approach as x gets closer and closer to the specified value, in this case 1.

What is the purpose of the "1-x" term in the equation?

The "1-x" term is used to represent the difference between the value of x and the limit value of 1. This allows us to evaluate the limit of the function as x gets closer to 1, rather than at the exact value of 1.

How is Summation by Parts used in real-world applications?

Summation by Parts is commonly used in physics and engineering to calculate the behavior of systems as they approach a certain value or limit. It is also used in economics and finance to model the growth or decline of variables over time.

What is the significance of the value L in the equation?

The value L represents the limit of the function as x approaches 1. It is the final result of the summation by parts method and gives us information about the behavior of the function at the specified limit.

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