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Determine whether the following functional series are pointwise convergent. In the case of convergence, show that the convergence is uniform or that it is not uniform, as the case may be.

$a)\sum_{j=0}^\infty\frac{\sin(jx)}{(2+x^2)^j}$

$b) \sum_{j=0}^\infty \frac{1}{(x+j)^2}$

Where $x\in\mathbb{R}$

I understand there are several well-known tests for convergence of a numerical series, e.g root test, comparison test and ratio test.

we can also use them in testing for uniform convergence together with the M-test.

The M-Test:

Let $f_j:D\to\mathbb{R}$ be bounded functions such that $||f_j||_{\infty}\leq a_j$ for some numbers $a_j$. If $\sum_{j=0}^\infty a_j$ is convergent, then $\sum_{j=0}^\infty f_j$ is uniformly convergent.

May sound stupid, I have all this information to use yet don't understand how to apply to these questions, hope someone can help, thanks in advance.

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  • $\begingroup$ What is the domain of your functions? $\endgroup$ – mechanodroid Jun 1 '18 at 15:59
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Hint:

$$\sum_{j=0}^\infty\frac{\left|\sin(jx)\right|}{(2+x^2)^j} \le \sum_{j=0}^\infty \frac{1}{(2+x^2)^j} \le \sum_{j=0}^\infty \frac{1}{2^j} = 2$$

so it converges uniformly on $\mathbb{R}$ by the M-test.

On the other hand the function $\sum_{j=0}^\infty \frac{1}{(x+j)^2}$ is not even well-defined e.g. for $x = -1$.

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