What are all the values of x for which the series below converges?

Question

What are all the values of x for which the series  converges?

	−1 ≤ x ≤ 1
	-1 ≤ x < 1
	-1 < x ≤ 1
	All reals

Answer 1

u_n = xⁿ/(4n²).

Using limit ratio test lim n→ ∞ Ix^{n + 1}/4(n + 1)²/Ixⁿ/4n²I = IxI < 1, so (-1, 1) is interval of convergence only we have to check x = - 1 and x = 1.

At x = 1 we have we have series ∑₁^∞ 1/4n² which is convergent as p-series with p 2 > 1.

AT x = -1 we have series ∑₁^∞ (-1)ⁿ/4n² _{which is absolutli converges.}

_{So, interval of convergence: -1 ≤ x ≤ 1}