- #1

Fernando Rios

- 96

- 10

- Homework Statement
- The following series are not power series, but you can transform each one into a power

series by a change of variable and so find out where it converges.

- Relevant Equations
- Σ((√(x^2+1))^n 2^n/(3^n+n^3))

We transform the series into a power series by a change of variable:

y = √(x

We have the following after substituting:

∑(2

We use the ratio test:

ρ

ρ = |(3

|2y| < 1

|y| = 1/2

-1/2 < y < 1/2

We find the possible values of "x":

-1/2 < √(x

This inequality has no solution, so I conclude the series doesn't converge for any "x", but the book says the answer is |x| < √(5)/2. Could you please tell me what am I doing wrong?

y = √(x

^{2}+1)We have the following after substituting:

∑(2

^{n}y^{n}/(3^{n}+n^{3}))We use the ratio test:

ρ

_{n}= |(2^{n+1}y^{n+1}/(3^{n+1}+(n+1)^{3})/(2^{n}y^{n}/(3^{n}+n^{3})| = |(3^{n}+n^{3})2y/(3^{n+1}+(n+1)^{3})|ρ = |(3

^{∞}+∞^{3})2y/(3^{∞+1}+(∞+1)^{3})| = |2y||2y| < 1

|y| = 1/2

-1/2 < y < 1/2

We find the possible values of "x":

-1/2 < √(x

^{2}+1) < 1/2This inequality has no solution, so I conclude the series doesn't converge for any "x", but the book says the answer is |x| < √(5)/2. Could you please tell me what am I doing wrong?