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I'm trying to find the power series representation for:

$$f(x)\;=\;\frac{3x^2+4x-5}{x(x+1)(x^2+x+1)}$$

without a reasonable solution.

Any help you could give me will be appreciated.

C. Ballez
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    About which point? – Alex Provost May 11 '17 at 23:31
  • The function has a pole at zero and is undefined there. Similarly, there are other poles coming from the roots of the denominator. However, if you allow Laurent series, you can express it as a power series but with an extra term for the pole. So, in this case you have $f(x ) = -5/x + 14 - 15x + ...$. – Somos May 11 '17 at 23:51
  • But there is one Laurent series for the region $0 < |x| < 1$ and a different one for $|x| > 1$. – Robert Israel May 12 '17 at 00:50

2 Answers2

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HINT Expand the function in partial fractions

$$ -\frac{x+2}{x^2+x+1}+\frac{5}{x}-\frac{4}{x+1} $$

And expand each term. For instance

$$ \frac{1}{x+1}=\sum_{k=0}^{+\infty}(-1)^k x^k ~~~\mbox{for}~~~ |x| < 1 $$

can you take it from here?

caverac
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Major components

This should be etched into memory: $$ \frac{1}{x+1} = 1 - x + x^{2} - x^{3} + x^{4} + \dots $$ Notice the missing powers are $2, 8, 14, \dots$ $$ \frac{1}{x^{2}+x+1} = 1 - x + x^{3} - x^{4} + x^{6} - x^{7} \dots $$

Assemble using multiplication

Missing powers are $3, 4, 9, 10, 15, 16, \dots$ $$ \frac{1}{x \left( x+1 \right)\left( x^{2}+x+1 \right)} = \frac{1}{x} - 2 + 2 x - x^{2} + x^{5} - 2x^{6} + 2x^{7} - x^{8} + x^{11} + \dots $$

Final answer: $$ \frac{3x^{2} + 4x - 5}{x \left( x+1 \right)\left( x^{2}+x+1 \right)} = -\frac{5}{x} +14 - 15 x+ 7x^{2} + 2x^{3} - 3x^{4} - 5x^{5} + 14x^{6} -15 x^{7} + \dots $$

plot

dantopa
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