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Represent the function \displaystyle \frac{9}{(1 - 5 x)^2} as a power series \displaystyle f(x) = \sum_{n=0}^\infty c_n x^n c_0 = c_1 = c_2 = c_3 = c_4 = Find the radius of convergence R = .

Solution

5 (1 Ratings )

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**Question**

Represent the function \displaystyle \frac{9}{(1 - 5 x)^2} as a power series \displaystyle f(x) = \sum_{n=0}^\infty c_n x^n c_0 = c_1 = c_2 = c_3 = c_4 = Find the radius of convergence R = .

** **

**Solution**

1/(1-x)=_{[n =0 to infinity]} x^{n}

1/(1-5x)=_{[n =0 to infinity]} (5x)^{n}

1/(1-5x)=_{[n =0 to infinity]} 5^{n}x^{n}

differentiate with respect to x

5/(1-5x)^{2}=_{[n =0 to infinity]} n5^{n}x^{n-1}

1/(1-5x)^{2}=_{[n =1 to infinity]} n5^{n-1}x^{n-1}

1/(1-5x)^{2}=_{[n =0 to infinity]} (n+1)5^{n}x^{n}

9/(1-5x)^{2}=_{[n =0 to infinity]} 9(n+1)5^{n}x^{n}

c0=9

c1=90

c2=675

c3=4500

c4=28125

a_{n}=9(n+1)5^{n}x^{n}

a_{n+1}=9(n+2)5^{n+1}x^{n+1}

for convergence lim_{n->infinity}|a_{n+1}/a_{n}|<1

lim_{n->infinity}|9(n+2)5^{n+1}x^{n+1} /[9(n+1)5^{n}x^{n}]|<1

lim_{n->infinity}|(1+(1/(n+1)))5x]|<1

|(1+(0))5x]|<1

|5x|<1

|x|<1/5

radius of convergence R =1/5

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