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I'm tasked with showing the following:

given the series [tex]\sum_{n=1} ^{\infty} \frac{1}{x^2 + n^2}[/tex]

(1) show that it converges Uniformly [tex]f_n(x) :\mathbb{R} \rightarrow \mathbb{R}[/tex].

(2) Next show the function

[tex]f(x) = \sum_{n=1} ^{\infty} \frac{1}{x^2 + n^2} [/tex]

is continious on [tex]\mathbb{R}[/tex]

(1) Suppose [tex]f_n = \frac{1}{x^2 + n^2}[/tex],

Then [tex]f_n[/tex] is uniformly convergens if

[tex]sup _{x \in \mathbb{R}} |f_n(x) - x|[/tex]. Now

[tex]_{sup} _{x \in \mathbb{R}} |f_n(x) - x| = |\frac{1}{x^2 - n^2} - x| =

_{sup} _{x \in \mathbb{R}} \frac{1}{x^2 - n^2}[/tex]

The deriative of f_n(x) is non-negative on [tex]\mathbb{R}[/tex], so its increasing and is hence maximumized at [tex]x = \mathbb{R}[/tex]. So the supremum is [tex]1/n^2[/tex]. This does tend to zero as [tex]n \rightarrow \infty[/tex]. So therefore it converge Uniformly.

Am I on the right track here? If yes any hints on how to prove the continuety ?

I know that its something to do with:

\integral_{1} ^{\infty} 1/x^2 + n^2 dx = \sum_{n=1} ^{\infty} \frac{1}{x^2 + n^2}

Sincerely Hummingbird25

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# Uniform convergens and continuity on R

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