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Addition, Multiplication, and Substitution of Taylor Series

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Explanations 1

There are several manipulations that you can perform on Taylor series, all of which are fairly easy to understand. First, we will add two Taylor series. To do this, simply add l…
By Sujay Kazi, Alex Federspiel

Problems 3

What's the coefficient of x10x^{10}x10 in the Maclaurin series of (1−3x+2x2)−1(1 - 3x + 2x^2)^{-1}(1−3x+2x2)−1? Source: HMMT February 2001
Let I=∫01ln⁡xln⁡(1−x) dxI = \int_0^1 \ln{x} \ln(1-x) \; dxI=∫01lnxln(1−x)dx. What is the value of I+∑n=1∞n−2I + \sum_{n=1}^{\infty} n^{-2}I+∑n=1∞n−2? Source: HMMT February 2008
Let PPP be the quadratic function such that P(0)=7P(0) = 7P(0)=7, P(1)=10P(1) = 10P(1)=10, and P(2)=25P(2) = 25P(2)=25. If aaa, bbb, and ccc are integers such that every positiv…