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In class we learned how to test the convergence of series and how to calculate the sums of arithmetic and geometric series (if they exist) but are there methods to actually calculate the values non-arithmetic, non-geometric series? Can the value of any series be calculated or only those of a certain type? And can it still be done if they're infinite? So my question is: are there any general methods for calculating the values of certain non-arithmetic/geometric series (references would be appreciated as well) [because it seemed like a lot of effort for Euler just to calculate the sum from n=1 to infinity of $\frac1{n^2}$]. Thanks in advance!

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No, there is no general way to calculate a "general series" because there are just so many series, they can be very involved. There are even very simple finite sum like $\sum_{i=1}^n 1/i$ that have no simple closed form.

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  • $\begingroup$ But are there techniques to calculate certain series? $\endgroup$ – Gabriel Jan 27 '15 at 2:09
  • $\begingroup$ @Gabriel there are techniques for calculating everything (not just series) for special cases. $\endgroup$ – Zero Jan 27 '15 at 2:10
  • $\begingroup$ so are there any techniques for the calculation of non-arithmetic, non-geometric series? $\endgroup$ – Gabriel Jan 27 '15 at 2:19
  • $\begingroup$ @Gabriel There are but not in general, for example look at the first paragraph of this wikipedia article en.wikipedia.org/wiki/Telescoping_series. You can also use other techniques like integration or derivation of series but there is no method for calculating anything in general in math. $\endgroup$ – Zero Jan 27 '15 at 2:23
  • $\begingroup$ All right I get that there is no general method for calculation, but are there ways to calculate certain classes of series, like is done with telescoping series? $\endgroup$ – Gabriel Jan 27 '15 at 2:33

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