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Unformatted text preview: es (we know the numerator is not changing, but the denominator is getting larger)
2. an is always positive (as n is always positive, n2+1 will always be positive)
3. f(n) = an is a continuous function (there is no discontinuity in this function at all) This means we can compare the series to the integral:
This means both the integral and the series converge by the Integral test! 1 (Note that the series does not dn ArcTan (n) 1
converge to this value, it simply n2 1
means it converges) 1 dn ArcTan () ArcTan (1) n2 1
1 Example 3 n
Consider the following series: 2
n 1 n 1
1. an decreases. We must find the derivative to show that the following function decreases eventually: n
f ( n) 2
(1)(n 2 1) (2n)(n)
f ' ( n) 2
n2 1 1 n2 f ' (...
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