Absolute convergence of a series and parameters

Question

I've been taught that if a series is absolutely convergent, then it is convergent.

I have a question, because my series has got a parameter within it. Let's call it b.

If a series is absolutely convergent for a parameter within an interval, I can conclude that in that interval it is also convergent (in my language, we call it "simply" convergent or convergent...does this nomenclature apply also for english? ...googled but found nothing...)

But my question is : could another interval of the parameter exist for which the series is also "simply" convergent?

The series I'm talking is

$$\sum_{n=1}^{\infty} (-1)^n \sin\left(\frac{1}{n}\right) \frac{(b^n + 6^n)}{7^n}$$

I think that for $$|b|<7$$ the series is absolutely convergent (and also convergent).

Are there other interval in which the series is convergent?

Thanks

Answer 1

One can apply the limit comparison test to see that if your series converges, then the following series must converge, and vice versa.

$$\sum_{n=1}^\infty\frac{(-1)^n}n\frac{b^n}{7^n}$$

since $\sin(1/n)\sim\frac1n$. From here, we note that by the Leibniz criterion, this series converges for $b=7$.

By the ratio test, it cannot converge for $|b|>7$.

By the integral test, it cannot converge for $b=-7$.

So the full set of solutions is $-7<b\le7$