Find a power series representation for the function and determine the radius of

vestirme4 2021-10-29 Answered
Find a power series representation for the function and determine the radius of convergence. f(x)=x(1+4x)2
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Expert Answer

Faiza Fuller
Answered 2021-10-30 Author has 108 answers

Step 1
We will start with the power series of 11x
We know that
11x=n=0xn
Step 1 : Replace x with 4x
11(4x)=n=0(4x)n
11+4x=n=0(4)nxn
Step 2
Differentiate both sides
ddx[11+4x]=ddx[n=0(4)nxn]
4(1+4x)2=n=0(4)nd[xn]dx
4(1+4x)2=n=1(4)n(n)xn1
Not that the sum in RHS is starting from n=1, because the first term died during differentiation
Divide both sides by 4, To get
1(1+4x)2=n=1(4)n1(n)xn1
1(1+4x)2=n=1(1)n14n1(n)xn1
Step 3 : Multiply both sides by x
x(1+4x)2=n=1(1)n1(4)n1(n)xn
If you want you can start the sum from n=0, but if you do that, you should replace n with n+1 in the sum

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