Does Any Non-Trivial Power Series Uniformly Converge on All R?

[WannaBe22]

Homework Statement

1.Is there any non-trivial power series that uniformly converges in all R? (a non-trivial power series has infinite non-zero coefficients...)

2. 2. Let f(X) be defined in [a,b]. We'll define fn(x) = [nf(x)] / n where [t]=floor value of t...
Check if the series (fn(x)) uniformly converges in [a,b].

Homework Equations

The Attempt at a Solution

I've no idea how to start solving these questions...I really need some guidance in those...

Thanks

 

[mighty2000]

for your post, and for your honesty about needing guidance on these questions. it is important to have a strong foundation in mathematical concepts and problem-solving skills. Let's break down these two questions and see how we can approach them step by step.

1. Is there any non-trivial power series that uniformly converges in all R?

A power series is a series of the form Σanxn, where an are the coefficients and xn are the powers of x. In order for a power series to converge, the limit of its terms must approach 0. In other words, as n approaches infinity, the value of anxn must approach 0.

Now, for a power series to uniformly converge in all R, it must converge at every point in the real number line, not just at a specific interval. This means that the convergence of the series must not depend on the value of x. In other words, the convergence must be uniform.

To determine if there is a non-trivial power series that uniformly converges in all R, we can use the ratio test. This test helps us determine the radius of convergence of a power series. If the radius of convergence is infinite, then the series will converge in all R.

2. Let f(X) be defined in [a,b]. We'll define fn(x) = [nf(x)] / n where [t]=floor value of t...

In this question, we are given a function f(x) defined in the interval [a,b]. We are then asked to define a new sequence fn(x) using the floor function [t], which rounds down a real number to the nearest integer. The goal is to determine if the series (fn(x)) uniformly converges in the interval [a,b].

To solve this, we can use the Weierstrass M-test. This test helps us determine if a series of functions uniformly converges by comparing it to a convergent series of constants. If the series of constants converges, then the original series of functions will also converge uniformly.

I hope this helps you get started on solving these questions. Remember to always break down the problem into smaller steps and use the appropriate tests and theorems to guide you. Good luck!