n_13534 - Calculus 2 Convergence of Series(Infinite Sums...

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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 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: a n = 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 n = n 0 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 infinite limit.
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