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Unformatted text preview: g the areas: f (n)dn a 1 1 a2 a3 a4 ...an 1 Integrals and Areas
n 1 f (n)dn a 1 a2 a3 a4 ...an 1 1 This means that our partial sum from 1 to n is:
n 1 f (n)dn S n 1 If the integral is continuous and diverges to infinity (everything is positive!) Sn
This means our sum is larger than infinity. The sequences diverges. Integral Test
Consider an infinite series that has the following properties:
1. an is decreasing (eventually)
2. an is always positive (eventually)
3. f(n) = an is a continuous function (eventually) f (n)dn
If diverges, then so does the series.
1 f (n)dn
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