Find the power series and interval of convergence for the following --> 1/(2+x)?

Question

Find the power series and interval of convergence for the following --> 1/(2+x)

Answer 1

1/(2+x) = 1/2* 1/(1+x/2).

The power series for 1/(1-q) is the geometric series 1+q+q^2+..., if abs(q)<1, so now you have to substitute q=-x/2 to get
1/(1+x/2) = 1-x/2+x^2/4-x^3/8+x^4/16-..., if abs(-x/2)=abs(x/2)<1, that is, if abs(x)<2. That means that the radius of convergence is 2.

Finally, you have to multiply the whole thing by 1/2, but this does not affect the radius of convergence (the sum of the series is finite if and only if 1/2 times the sum is finite).

Answer 2

Using Maclaurin's theorem,
1/((x+2) =1/2 - x/4 + x^2/8 - x^3/16 .....(-1)^n*(x^n)/2^(n+1)
Using the ratio test, the interval of convergence is |x/2|<1, or
-1< x/2 < 1 or -2 < x < 2.

You can now plug in x =-2 and see if the resulting series is convergent or not. If it is, the interval of convergence includes the left end point. Do the same to see if the right end point is included.