If $\sum\limits_{0}^{\infty}{5^n.a_n}$ converges and $\sum\limits_{0}^{\infty}{(-5)^n.a_n}$ diverges, choose the correct answer

Question

I found an interesting problem, which I'm unable to solve completely:

If $\sum\limits_{0}^{\infty}{5^n.a_n}$ converges and $\sum\limits_{0}^{\infty}{(-5)^n.a_n}$ diverges, choose the correct answer:

$\sum\limits_{0}^{\infty}{5^n.a_n}$ converges: A) absolutely, B) conditionally

$\sum\limits_{0}^{\infty}{4^n.a_n}$ converges: A) absolutely, B) conditionally, C) does not

$\sum\limits_{0}^{\infty}{7^n.a_n}$ converges: A) absolutely, B) conditionally, C) does not

Now, $\sum\limits_{0}^{\infty}{|(-5)^n.a_n|}=\sum\limits_{0}^{\infty}{|5^n.a_n|}$. If $\sum\limits_{0}^{\infty}{5^n.a_n}$ converges absolutely, then $\sum\limits_{0}^{\infty}{(-5)^n.a_n}$ would also converge absolutely, but the task says it diverges. Therefore, $\sum\limits_{0}^{\infty}{5^n.a_n}$ converges conditionally (B). But what about the second and third bullet points? I have no idea what to do there.

@Mikasa Infortunately, what I've written is the only information available. — Kotaka Danski, Jul 08 '21 at 09:28
@blamethelag I can't think of a way to use the first bullet point to solve the other two. — Kotaka Danski, Jul 08 '21 at 09:29

score 3 · Accepted Answer · answered Jul 08 '21 at 09:30

3

The key result is Theorem 1 of this answer, the relevant part reads:

Theorem 1. Any power series $\sum_{n=0}^\infty a_n x^n $ that converges at one $x_0$ where $|x_0|=\rho>0$, converges absolutely and locally uniformly on the set $|x|<\rho $.

From this, we see immediately that it converges absolutely at $x=4$, And it cannot converge at $x=7$ (as this would imply the absolute convergence at $x=5$.)

answered Jul 08 '21 at 09:30

Calvin Khor

34,903

Thanks a lot! It turns out the answer is simple. – Kotaka Danski Jul 08 '21 at 09:53

If $\sum\limits_{0}^{\infty}{5^n.a_n}$ converges and $\sum\limits_{0}^{\infty}{(-5)^n.a_n}$ diverges, choose the correct answer

1 Answers1