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I've been reading my analysis notes. When they define the sequences of functions and the pointwise and uniform convergence, it says that it has to converge to a function $f$. Then, they define the series of functions and the statements are almost analogue to the sequences of functions, but here it says: a series of functions $\sum{f_n}$ converges pointwise on the interval $I$ if the sequence of partial sums converge pointwise. The same for the uniform convergence. After that, it's like they always say "this series converges uniformly here" but they don't say where, they don't say "it converges here to this function".

Why they do this? The natural way to see it invites to say that something converges to something. It's not necessary to check the uniform convergence of a function to know where it actually converges?

Thanks for taking the time.

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  • $\begingroup$ You can prove the convergence withouth knowing the limit. $\endgroup$ – Martín-Blas Pérez Pinilla Dec 31 '15 at 17:30
  • $\begingroup$ But wouldn't it be more "natural", on the definitions and propositions, to say: this series converges here to this function, instead of only saying that it converges on an interval? $\endgroup$ – Relure Dec 31 '15 at 17:33
  • $\begingroup$ In fact, determinating where converges the sequence/series is a very common problem. $\endgroup$ – Martín-Blas Pérez Pinilla Dec 31 '15 at 17:40
  • $\begingroup$ It is implicit in that if it converges at a point, then the limit is $\sum_{n=0}^\infty f_n(x)$. So there is no need to use another label (although it might make things more transparent). $\endgroup$ – copper.hat Dec 31 '15 at 17:47
  • $\begingroup$ Recall that the definition of convergence that you first learned, say $\lim s_n=s$, very explicitly required the value of the limit $s$ in the definition. Eventually you learn that convergence is an intrinsic property of the sequence and can be described without reference to the actual limit. Same here. We say "$f_n \to f$ on some set" or simply "$f_n$ converges on some set." This is more comfortable language for us and you just have to get used to it. (On some other planet they may well always refer to the limit when they discuss convergence.) $\endgroup$ – B. S. Thomson Dec 31 '15 at 17:55

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