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Find power series representation of the function $f(x) = \frac{3}{x+2}$

\begin{align*}f(x) = \left(\frac{3}{x}\right)\frac{1}{1-\left(-\frac{2}{x}\right)} = \left(\frac{3}{x}\right) \sum{\left(-\frac{2}{x}\right)}^{n}\end{align*}

Markus Scheuer
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Karthick B
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  • It is right! Just a hint, learn a few about latex – L.F. Cavenaghi Mar 17 '16 at 21:57
  • They probably mean a power series in $x$. Similar strategy, $\frac{3}{2}\cdot \frac{1}{1+x/2}$. – André Nicolas Mar 17 '16 at 21:58
  • In the definition of power series that I know, negative exponents of $x$ are not allowed. So, technically, your answer $f(x)=\frac{3}{x} \sum_{n=0}^\infty \left(\frac{-2}{x}\right)^n$ would not be OK. @André Nicolas has a correct answer. – Darío G Mar 17 '16 at 22:01

2 Answers2

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The usual meaning of power series is an expression of the form $$c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+\cdots.$$ Your answer is not of that form.

In your problem, $a$ has unfortunately not been specified. The default assumption is that $a=0$.

Express your function as $$\frac{3}{2}\cdot \frac{1}{1+x/2},$$ and use the same strategy as in the OP. You are probably expected to specify where your series converges.

André Nicolas
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What you obtained $$\frac 3 x \sum_{n=0}^\infty{\left(-\frac{2}{x}\right)}^{n}$$ is the perfect series representation of $f(x) = \frac{3}{x+2}$ for infinitely large values of $x$.

For illustration, let us just consider the case of $x=10$ : the exact value is $\frac 14$ while the partial sum using five terms would be $\frac{1563}{6250}\approx 0.25008 $

Claude Leibovici
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