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mote1985 [20]
4 years ago
8

Find a power series representation for the function. Determine the interval of convergence. (Give your power series representati

on centered at x = 0.) f(x) = 5 2 − x
Mathematics
1 answer:
kaheart [24]4 years ago
6 0

I'm guessing the function is

f(x)=\dfrac5{2-x}=5(2-x)^{-1}

whose first few derivatives are

\dfrac{\mathrm df}{\mathrm dx}=-5(2-x)^{-2}(-1)=5(2-x)^{-2}

\dfrac{\mathrm d^2f}{\mathrm dx^2}=-2\cdot5(2-x)^{-3}(-1)=2!\cdot5(2-x)^{-3}

with a general pattern of

\dfrac{\mathrm d^nf}{\mathrm dx^n}=n!\cdot5(2-x)^{-(n+1)}=\dfrac{5n!}{(2-x)^{n+1}}

Then f has power series (centered at 0)

\displaystyle\sum_{n=0}^\infty\frac{\frac{5n!}{2^{n+1}}}{n!}x^n=\frac52\sum_{n=0}^\infty\left(\frac x2\right)^n

which is a geometric series, converging only for

\left|\dfrac x2\right|

(this is the interval of convergence)

###

If you know that for |x|

\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n

then we can derive the same series by manipulating this result:

\displaystyle\frac5{2-x}=\frac52\frac1{1-\frac x2}=\frac52\sum_{n=0}^\infty\left(\frac x2\right)^n

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