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Unformatted text preview: Calculus 2: Convergence of Series (Infinite Sums) Often the value of a series (the sum of all terms in an infinite sequence ) cannot be determined 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 approximate numerical values for various quantities. A few famous examples are e = 1 + 1 + 1 / 2 + 1 / 3! + ··· + 1 /n ! + ··· π = 4- 4 / 3 + 4 / 5- 4 / 7 + ··· + 4(- 1) n +1 2 n- 1 + ··· For the first of these examples, the eleventh partial sum s 11 = 1 + 1 + 1 / 2 + 1 / 6 + ··· + 1 / 10! = 2 . 718281801 ... which approximates e = 2 . 718281828 ... correct up to the seventh decimal place. However, to use such approximations, we first must check whether the series con- verges , which cannot be determined just by looking at partial sums. For example, the sum of the first billion terms for the harmonic series ∞ X n =1 1 /n = 1 + 1 / 2 + 1 / 3 + ··· is less 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 is done by a calculator of computer. There are a number of tests, each being the best for 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 order to try them in. I will describe the tests for series with the index starting at one: X a n = ∞ X n =1 a n However, remember that you can always ignore or change some terms at the beginning of a sequence without changing its convergence, so these tests all work for the general case ∞ X n = n a n . For sequences with no negative terms, a n ≥ 0, either the sum is finite or the partial sums go to infinity and so the series diverges. In the latter case I will say that ∑ a n = ∞ , but as usual be careful with infinity which is not a number, and never say this for divergence of other series where the partial sums might not have even an...
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