Find the Interval of Convergence of this Power Series: ∑(x^2n/(2^nn^2))

[Fernando Rios]

Homework Statement

Find the interval of convergence of each of the following power series; be sure to investigate
the endpoints of the interval in each case.

Relevant Equations

∑(x^(2n)/((2^n)(n^2)))

∑(x²ⁿ/(2ⁿn²))

We use the ratio test:
ρ_n = |(x²n²/(2(n+1)²)|

ρ = |x²/2|

ρ < 1

|x²| < 2

|x| = √(2)

We investigate the endpoints:
x = 2:
∑(4ⁿ/(2ⁿn²) = ∑(2ⁿ/n²))

We use the preliminary test:
lim_n→∞ 2ⁿ/n² = ∞

Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be included. However, then answer says it should be included.

 

[Mark44]

Homework Statement:: Find the interval of convergence of each of the following power series; be sure to investigate
the endpoints of the interval in each case.
Relevant Equations:: ∑(x^(2n)/((2^n)(n^2)))

∑(x²ⁿ/(2ⁿn²))

We use the ratio test:
ρ_n = |(x²n²/(2(n+1)²)|

ρ = |x²/2|

ρ < 1

|x²| < 2

|x| = √(2)

We investigate the endpoints:
x = 2:

This isn't one of the endpoints -- they are $-\sqrt 2$ and $\sqrt 2$.

∑(4ⁿ/(2ⁿn²) = ∑(2ⁿ/n²))

We use the preliminary test:
lim_n→∞ 2ⁿ/n² = ∞

Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be included. However, then answer says it should be included.

 

[Fernando Rios]

This isn't one of the endpoints -- they are $-\sqrt 2$ and $\sqrt 2$.

Thank your for your answer.