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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

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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 $-7<b\le7$

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  • $\begingroup$ Sorry, i mistyped my series ;) I updated the question, can you look at it please? $\endgroup$ – Jacquelyn.Marquardt Jan 6 '17 at 21:36
  • $\begingroup$ @f126ck Oh, well I'll see to that one then. $\endgroup$ – Simply Beautiful Art Jan 6 '17 at 21:37
  • $\begingroup$ Thanks and again sorry for that sir. $\endgroup$ – Jacquelyn.Marquardt Jan 6 '17 at 21:42
  • $\begingroup$ @f126ck No problem, I'm done. $\endgroup$ – Simply Beautiful Art Jan 6 '17 at 21:42
  • $\begingroup$ Thank you very much, I'm accepting the answer. Let me see if I understood. The series is ABSOLUTELY convergent for |b|<7 and for Leibniz it is "simply" convergent for |b|<7 AND 7 . So back to my question, It's true that there CAN be an interval of the parameter (in this series it is only a point, b=7) in which the series is convergent but NOT ABSOLUTELY convergent $\endgroup$ – Jacquelyn.Marquardt Jan 6 '17 at 21:54

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