Math  /  Calculus

QuestionRepresent the function 10(17x)\frac{10}{(1-7 x)} as a power series f(x)=n=0cnxnf(x)=\sum_{n=0}^{\infty} c_{n} x^{n} c0=c1=c2=c3=c4=\begin{array}{l} c_{0}=\square \\ c_{1}=\square \\ c_{2}=\square \\ c_{3}=\square \\ c_{4}=\square \end{array}
Find the radius of convergence R=R= \square .

Studdy Solution
Determine the radius of convergence RR for the series n=0(7x)n\sum_{n=0}^{\infty} (7x)^n.
The radius of convergence for a geometric series 11ax\frac{1}{1-ax} is given by ax<1|ax| < 1.
7x<1 |7x| < 1 x<17 |x| < \frac{1}{7}
Thus, the radius of convergence R=17R = \frac{1}{7}.
The power series representation is:
f(x)=n=0107nxn f(x) = \sum_{n=0}^{\infty} 10 \cdot 7^n x^n
With the coefficients:
c0=10,c1=70,c2=490,c3=3430,c4=24010 c_0 = 10, \quad c_1 = 70, \quad c_2 = 490, \quad c_3 = 3430, \quad c_4 = 24010
And the radius of convergence:
R=17 R = \frac{1}{7}

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