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Can someone explain how I should be solving for theses terms? I got my series to be $10(-7x)^n$, but I dont know what to do from there.

Represent the function $f(x)=\frac{10}{(1-7x)}$ as a power series $$f(x)=\sum_{n=0}^\infty c_nx^n$$

Find the following coefficients:

$c_0=\color{green}{10(-7(2))^0}\\c_1=\\c_2=\\c_3=\\c_4=$

Find the radius of convergence

$R=$

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  • $\begingroup$ Where does that "2" come from, in $10(-7(2))^0$? $\endgroup$
    – Gerry Myerson
    Apr 22, 2015 at 12:37
  • $\begingroup$ Use the binomial series for $g(y)=(1-y)^{-1}$ with $y=7x$ and them multiply through by $10$. The radius of convergence is then inherited from that of the series for $g(y)$ $\endgroup$
    – Conrad Turner
    Apr 22, 2015 at 13:10

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I think that in the preamble of your question you should have $10(7x)^n$ instead of the negative (this from Geometric series). It should be clear then that $c_n=10\cdot 7^n$. As far as the radius of convergence you should know that in the Geometric series, $|7x|<1$.

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