I of course read the Wikipedia article, but it sounded like such an abstract idea, that I could interpret only as a limit.

The term appeared in a recent lecture apparently out of nowhere (though I was slightly late) - an example the lecturer has now given online makes it seem just like a limit to me (found the radius of convergence by doing a ratio test) but I wonder if there is some distinction in the terminologies, otherwise one wonders why anyone would favour the term over simply 'limit' - perhaps it is a series vs sequence distinction?


1 Answer 1


They are different things.

Consider a series like $1+2x+(2x)^2+\ldots$. This is said to "converge" if the sequence of partial sums ($s_1=1,s_2=1+2x,\ldots$) converges (has a limit) as a sequence. There are some values of $x$ for which it does, like $x=1/4$, and some for which it doesn't, like $x=10$. The interval of all such $x$ in this case happens to be $(-1/2,1/2)$, which has half-length, or "radius" $1/2$. (Also, in the complex plane, the values of $x$ form a 2D disc with a more conventional "radius".)

We say that the radius of convergence is $1/2$ to mean that "the set of $x$-values for which the sequence of partial sums converges to a limit at all has radius $1/2$." This is separate from the fact that, when it does converge, the sequence of partial sums has limit $1/(1-2x)$.

  • $\begingroup$ So is the term used only for sequences of partial sums, or to describe a condition on the convergence of any series? $\endgroup$
    – OJFord
    Jan 27, 2014 at 1:49
  • $\begingroup$ @OllieFord, "convergence of a series" is defined to be the same as "convergence of the sequence of partial sums". The phrase "radius of convergence" is related to convergence of a series depending on a parameter. Equivalently, the phrase "radius of convergence" is related to convergence of a sequence of partial sums depending on a parameter". $\endgroup$
    – Mark S.
    Jan 27, 2014 at 1:52
  • $\begingroup$ I think I may also be confused about sequence vs series, my understanding was that sequences are finite sets with no necessary relationship between constituent members, e.g. ${2x, 7x, 54x}$, and series may be finite or infinite, but have a defining relations hip, like $\sum 2nx$. But then the sequence of partial sums you mention is certainly infinite; I don't see how it differs from the series it describes? $\endgroup$
    – OJFord
    Jan 27, 2014 at 1:53
  • 1
    $\begingroup$ @OllieFord A sequence can be finite or infinite, but is often infinite. A "series" is just adding up terms of a sequence. If you have $a_n=x*2n$ for every $n$ (starting at $0$), then the infinite sequence is something like $(0,2x,4x,6x,\ldots)$, the corresponding infinite series would be $0+2x+4x+6x+\ldots$, and the "sequence of partial sums" for that infinite series would be $(0,0+2x,0+2x+4x,0+2x+4x+6x,\ldots)$. Does that clear things up? $\endgroup$
    – Mark S.
    Jan 27, 2014 at 1:58
  • $\begingroup$ Enormously, thank you. Does that not also give $p_n = \sum a_n$ for every $n$ as a 'series of partial sums' for the original series? I see that convergence in one would be necessary and sufficient for the other, but the limit not (necessarily) equal, surely? $\endgroup$
    – OJFord
    Jan 27, 2014 at 2:04

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