Can someone help with this algebraic manipulation involving series?

topsquark

Math Team
May 2013
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The Astral plane
Work out a few terms and we can see a pattern.
\(\displaystyle c_{n + 2} = \frac{n - 1}{n - 2}c_n \text{ , } c_0 = -\frac{1}{2}c_0 \text{ , }c_3 = 0\)

We can immediately say that all \(\displaystyle c_n = 0\) for odd n > 1.

Now, \(\displaystyle c_4 = \frac{4 - 1}{4 + 2} = \frac{3}{6} c_2 = -\frac{1}{2} \cdot \frac{3}{6} c_0\)

Similarly for all even n > 2:
\(\displaystyle c_n = -\frac{1}{2} \cdot \frac{3}{6} \cdot \frac{5}{8} \cdot \text{ ... } \cdot \frac{n - 1}{n + 2} c_0\)

Thus
\(\displaystyle y(x) = c_0 + c_1 x - \frac{1}{2}c_0 x^2 - \frac{1}{2} \cdot \frac{3}{6} c_0 x^4 - \text{ ...}\)

-Dan
 
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