Weierstrass M-Test: Show Uniform Convergence on -infinity<x<infinity

[math&science]

How do I show that Sigma from 3 to infinity of 1/(n^2+x^2) is uniformly convergent on -infinity< x<infinity using the M-test? Can anyone help? Thanks in advance.

 

[Galileo]

How do I show that Sigma from 3 to infinity of 1/(n^2+x^2) is uniformly convergent on -infinity< x<infinity using the M-test? Can anyone help? Thanks in advance.

Well, you need to find terms $M_n$ with $|1/(n^2+x^2)|\leq M_n$ for all x, such that:
$$\sum_{n=3}^{\infty}M_n$$ is convergent.

Looking at your function, does any series come to mind?

 

[math&science]

1/n^2? That's what I thought of initially. Is that right and that simple?

 

[Galileo]

Why the doubt?
Is $1/(n^2+x^2)<br /> \leq 1/n^2$?
Is $\sum_{n=3}^{\infty} 1/n^2$ convergent? If so, then according to the M-test your series is uniformly convergent. It's that simple.