# If a Power Series is known to converge at a point, what can we conclude?

If it is known that the series $$\sum_{n=1}^{\infty} a_nx^n$$ is convergent at $$x=4$$. What can we conclude about the series $$\sum_{n=1}^{\infty} a_n(-7)^n$$?

A. Convergent
B. Conditionally Convergent
C. Conditionally Convergent
D. Divergent
E. May be convergent or divergent

Is my logic right? Since we are told that the series is convergent at $$x=4$$, then this might be a point inside the convergence interval or one of the endpoints, however, we do not know. Hence, the series $$\sum_{n=1}^{\infty} a_n(-7)^n$$ might be convergent or divergent (option $$E$$ is right).

• E is correct. Your logic is essentially right; however, it would be good to illustrate that it can be convergent with an example, and divergent with an example in order to be totally sure. For divergent, consider $a_i = \frac{1}{(-7)^i}$. For convergent, consider $a_i = 0$. Aug 20, 2020 at 19:42
• @user62487108 A minor point is your option B and C are identical, so I assume it's a typo. Aug 20, 2020 at 21:15

You chose the right option, but you should explain why it may converge or diverge. For instance, both series $$\sum_{n=0}^\infty\frac{x^n}{5^n}$$ and $$\sum_{n=0}^\infty\frac{x^n}{8^n}$$ converge when $$x=4$$. However, the first one diverges when $$x=-7$$, whereas the second one converges.