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Sidana [21]
2 years ago
5

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do n

ot show that Rn(x) → 0.] f(x) = 5(1 − x)−2 f(x) = ∞ n = 0 Find the associated radius of convergence R. R =
Mathematics
1 answer:
Mazyrski [523]2 years ago
5 0

We can use the fact that, for |x|,

\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n

Notice that

\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1{1-x}\right]=\dfrac1{(1-x)^2}

so that

f(x)=\displaystyle\frac5{(1-x)^2}=5\frac{\mathrm d}{\mathrm dx}\left[\sum_{n=0}^\infty x^n\right]

f(x)=\displaystyle5\sum_{n=0}^\infty nx^{n-1}

f(x)=\displaystyle5\sum_{n=1}^\infty nx^{n-1}

f(x)=\displaystyle5\sum_{n=0}^\infty(n+1)x^n

By the ratio test, this series converges if

\displaystyle\lim_{n\to\infty}\left|\frac{(n+2)x^{n+1}}{(n+1)x^n}\right|=|x|\lim_{n\to\infty}\frac{n+2}{n+1}=|x|

so the series has radius of convergence R=1.

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