Use differentiation to find a power series representation for f(x) = 1/(5 + x)^2

Question

Use differentiation to find a power series representation for f(x) = 1/(5 + x)^2. What is the radius of convergence, R? Use part (a) to find a power series for f(x) = 1/(5 + x)3 What is the radius of convergence, R? Use part (b) to find a power series for f(x) = x^2(5 + x)^3 What is the radius of convergence, R? Please try again, keeping in mind that you can differentiate both sides of the power series expansion to obtain power series representation for the derivative of the given function. So, to find the power series representation of a function of the form a / (b + cx)^2, you may first find the power series representation of the function a / b + cx. The resulting power series converges on the same interval.

Explanation / Answer

a)1/(5+x)=(1/5)(1/(1+(x/5)))=(1/5)(1/(1-(-x/5)))

=(1/5) sumn=0 to infinity(-x/5)n

=(1/5) sumn=0 to infinity(-1)n(x/5)n

= sumn=0 to infinity(-1)n(1/5)n+1xn

1/(5+x)= sumn=0 to infinity(-1)n(1/5)n+1xn

differentiate with respect to x

-1/(5+x)2= sumn=1 to infinity(-1)n(1/5)n+1nxn-1

1/(5+x)2= sumn=1 to infinity(-1)n+1(1/5)n+1nxn-1

1/(5+x)2= sumn=0 to infinity(-1)n+2(1/5)n+2(n+1)xn

radius of convergence =5

b)1/(5+x)2= sumn=0 to infinity(-1)n+2(1/5)n+2(n+1)xn

differentiate with respect to x

-2/(5+x)3= sumn=0 to infinity(-1)n+2(1/5)n+2(n+1)nxn-1

2/(5+x)3= sumn=0 to infinity(-1)n+1(1/5)n+2(n+1)nxn-1

1/(5+x)3= sumn=1 to infinity(-1)n+1(1/2)(1/5)n+2(n+1)nxn-1

1/(5+x)3= sumn=0 to infinity(-1)n(1/2)(1/5)n+3(n+2)(n+1)xn

radius of convergence =5

c)x2/(5+x)3= sumn=0 to infinity(-1)n(1/2)(1/5)n+3(n+2)(n+1)xn*x2

x2/(5+x)3= sumn=0 to infinity(-1)n(1/2)(1/5)n+3(n+2)(n+1)xn+2

radius of convergence =5

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.