Calculus 2: Convergence of Series (Infinite Sums)Often the value of aseries(the sum of all terms in an infinitesequence) cannot bedetermined exactly: the geometric series is one of the few that we can evaluate so far.However, partial sums of series can be a very useful way of computing approximatenumerical values for various quantities. A few famous examples aree=1 + 1 + 1/2 + 1/3! +· · ·+ 1/n! +· · ·π=4-4/3 + 4/5-4/7 +· · ·+4(-1)n+12n-1+· · ·For the first of these examples, the eleventh partial sums11= 1 + 1 + 1/2 + 1/6 +· · ·+ 1/10! = 2.718281801...which approximatese= 2.718281828...correct up to the seventh decimal place.However, to use such approximations, we first must check whether the seriescon-verges, which cannot be determined just by looking at partial sums. For example, thesum of the first billion terms for the harmonic series∞n=11/n= 1 + 1/2 + 1/3 +· · ·isless than thirty, giving no clear evidence that the whole sum is “infinity”.Thus “testing for convergence” (determining whether a series converges or di-verges) is often the main mathematical task in using a series: the rest of the work isdone by a calculator of computer. There are a number of tests, each being the bestfor some cases but not for all, so that one might have to try several to get an answer.Here I will summarize the tests and give some suggestions as to which orderto try them in.I will describe the tests for series with the index starting at one:an=∞n=1anHowever, remember that you can always ignore or change some terms at thebeginning of a sequence without changing its convergence, so these tests all work forthe general case∞n=n0an.For sequences with no negative terms,an≥0, either the sum is finite or thepartial sums go to infinity and so the series diverges. In the latter case I will say that∑an=∞, but as usual be careful with infinity which is not a number, and neversay this for divergence of other series where the partial sums might not have even aninfinite limit.