If we have $\sum (-1)^nx_n$ and if $x_n>0$, and $\lim_{n\rightarrow +\infty }x_n=0$ and $x_n$ is a decreasing sequence then

$\left | \sum_{n=q+1}^{p} (-1)^nx_n\right |\leq x_{q+1}$

And if $\underset{x}{sup}(x_{q+1})\rightarrow0 $ then the series converges uniformly

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    $\begingroup$ the first statement is true. The second makes no sense, what do you mean by supremum over all $x$'s? $\endgroup$ – Yanko Mar 30 at 17:21
  • $\begingroup$ To prove uniform convergence, I took supremum of both sides, if the R.H.S tends to zero ,then we're finshed. $\endgroup$ – Pedro Alvarès Mar 30 at 17:52
  • $\begingroup$ Maybe I'm wrong, it should be $x_{q+1}$ that tends to zero ? $\endgroup$ – Pedro Alvarès Mar 30 at 17:53
  • $\begingroup$ It is given that $x_n$ goes to zero. $\endgroup$ – Yanko Mar 30 at 18:01
  • $\begingroup$ Oh okay, but the thing I didn't understad is why $\left | \sum_{n=q+1}^{p} (-1)^nx_n\right |\leq x_{q+1}$ $\endgroup$ – Pedro Alvarès Mar 30 at 18:33

Presumably $x_n$ is a function of $x$ in some domain $D$ as you are concerned with uniform convergence.

Since $x_n$ is a decreasing sequence, $x_{n} - x_{n+1} > 0$ and

$$\left | \sum_{n=q+1}^{p} (-1)^nx_n\right| = \left |(-1)^{q+1}(x_{q+1} - x_{q+2} + x_{q+3} - x_{q+4} + \ldots) \right| \\= x_{q+1} - (\, x_{q+2} - x_{q+3}\, ) - (\, x_{q+3} - x_{q+4}\, ) - \ldots \leqslant x_{q+1} \leqslant \sup_{x \in D} x_{q+1} $$

Thus, if the RHS tends to $0$ as $q \to \infty$ then uniform convergence follows as the uniform Cauchy criterion is satisfied.


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