Please answer this calculus question?

Question

Let f be the function defined by f(x)=∑ 2((x+2)/3)^n from n=0 to infinity for all values of x for which the series converges. 

 

a) Find the interval of convergence for the series. 

b) Find the function that the series represents. 

 

Please show all your work. Thanks.

Answer 1

Let f be the function defined by f(x)=∑ 2((x+2)/3)^n from n=0 to infinity for all values of x for which the series converges.

a) Find the interval of convergence for the series.

((x+2)/3)^n < 1

(x+2)^n < 3^n

x+2 < 3

x < 1

Series converges for x on the interval (-∞, 1).

b) Find the function that the series represents.

f(x)=∑{n=0,∞}(2((x+2)/3)^n)

f(x)/2 = 1 + ((x+2)/3) + ((x+2)/3)^2 + …

((x+2)/3) f(x)/2 = ((x+2)/3) + ((x+2)/3)^2 + …

(1 – (x+2)/3)f(x)/2 = 1

f(x) = 2/(1 – (x+2)/3) = 6/(3 – (x+2))

f(x) = 6/(5 – x), x < 1