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Math  /  Calculus

QuestionRepresent the function 10(1−7x)\frac{10}{(1-7 x)} as a power series f(x)=∑n=0∞cnxnf(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

STEP 1

1. The function is of the form 11−ax\frac{1}{1-ax} which can be expanded using the geometric series formula.
2. The geometric series formula 11−x=∑n=0∞xn\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n is applicable here.
3. The power series representation will be valid within the radius of convergence.

STEP 2

1. Rewrite the given function in a form suitable for geometric series expansion.
2. Identify the coefficients cnc_n for the power series.
3. Determine the radius of convergence RR.

STEP 3

Rewrite the function 101−7x\frac{10}{1-7x} in the form 11−ax\frac{1}{1-ax} where a=7a = 7.
101−7x=10×11−7x \frac{10}{1-7x} = 10 \times \frac{1}{1-7x}
This is now ready for geometric series expansion.

STEP 4

Expand 11−7x\frac{1}{1-7x} using the geometric series formula:
11−7x=∑n=0∞(7x)n=∑n=0∞7nxn \frac{1}{1-7x} = \sum_{n=0}^{\infty} (7x)^n = \sum_{n=0}^{\infty} 7^n x^n
Multiply the entire series by 10:
10×∑n=0∞7nxn=∑n=0∞10⋅7nxn 10 \times \sum_{n=0}^{\infty} 7^n x^n = \sum_{n=0}^{\infty} 10 \cdot 7^n x^n
Thus, the coefficients cnc_n are 10⋅7n10 \cdot 7^n.

STEP 5

Identify the coefficients:
c0=10⋅70=10 c_0 = 10 \cdot 7^0 = 10 c1=10⋅71=70 c_1 = 10 \cdot 7^1 = 70 c2=10⋅72=490 c_2 = 10 \cdot 7^2 = 490 c3=10⋅73=3430 c_3 = 10 \cdot 7^3 = 3430 c4=10⋅74=24010 c_4 = 10 \cdot 7^4 = 24010

STEP 6

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 11−ax\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=0∞10⋅7nxn 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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