What are all values of x for which it converges?

Question

∞

What are all values of x for which ∑ (2ⁿxⁿ)/n converges?

n=1

a) x = -1/2 and 1/2

b) -1/2 ≤ x ≤ 1/2

c) -1/2 < x < 1/2

d) -1/2 < x ≤ 1/2

e) -1/2 ≤ x < 1/2

Answer 1

Use the Ratio Test.

[2ⁿ⁺¹ lxl ⁿ⁺¹) / (n+1)] ÷ [2ⁿ lxl ⁿ / n] = 2(n/(n+1)) lxl

Taking the limit as n approaches infinity of the above expression we get 2 lxl.

By the Ratio Test the series converges as long as 2 lxl < 1.

So, lxl < 1/2. This means that -1/2 < x < 1/2

Check the endpoints:

When x = 1/2, we get ∑(1/n) which is the harmonic series (divergent).

When x = -1/2, we have ∑(-1)ⁿ/n. This series converges by the Alternating Series Test.

Interval of convergence is -1/2 ≤ x <1/2. (Choice e)