#MathsInMinutes Day 41: Convergent series
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The sum of an ordered list of numbers is convergent if it tends towards a specific value or limit. Intuitively, we might imagine that a series settles down if the difference between successive partial sums, the series totalled to a specific number of terms, gets smaller and smaller. For example, if the sequence of partial sums is (1, S1, S2, S3, …), where
Sn= 1 + ½ + 1/3 + … + 1/n.
Then the difference between Sn and Sn+1 is 1/n+1. As n gets very large 1/1+n gets very small. But is this really enough to say that this series, known as the harmonic series, actually settles down to a limit?
It turns out that Sn in this case does not settle down, and the series is divergent. So although successive differences may get small, as with a convergent Cauchy sequence, this on its own is not enough to guarantee that series converges.
source:
https://proxy.duckduckgo.com/iu/?u=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2F2%2F22%2FHarmonicNumbers.svg%2F1200px-HarmonicNumbers.svg.png&f=1
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