Let $f(x) = \displaystyle \frac{3+6x}{x}$.

Compute

$f(x)$

$=$

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$f(1)$

$=$

$f^\prime(x)$

$=$

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$f^\prime(1)$

$=$

$f^{\prime\prime}(x)$

$=$

$\hskip 30pt$

$f^{\prime\prime}(1)$

$=$

$f^{\prime\prime\prime}(x)$

$=$

$\hskip 30pt$

$f^{\prime\prime\prime}(1)$

$=$

$f^{(iv)}(x)$

$=$

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$f^{(iv)}(1)$

$=$

$f^{(v)}(x)$

$=$

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$f^{(v)}(1)$

$=$

We see that the first term does not fit a pattern, but we also see that $f^{(k)}(1)$ = for $k\geq 1$.
Hence we see that the Taylor series for $f$ centered at $1$ is given by

$f(x) = 9+ \displaystyle \sum_{k=1}^\infty$ $(x - 1)^k$.

Hint: