Does (-2)^n/( 3^n+1) converge or diverge?

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07-04-2017
Mathematics

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Does (-2)^n/( 3^n+1) converge or diverge?

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LammettHash LammettHash · Answer 1 · 2017-04-07T16:46:39+02:00

[tex]\left|\dfrac{(-2)^n}{3^n+1}\right|=\dfrac{2^n}{3^n+1}<\dfrac{2^n}{3^n}=\left(\dfrac23\right)^n[/tex]

As [tex]n\to\infty[/tex], the sequence [tex]a_n=\left(\dfrac23\right)^n[/tex] converges to zero.

If you're talking about the infinite series

[tex]\displaystyle\sum_{n\ge0}\dfrac{(-2)^n}{3^n+1}[/tex]

well we've shown by comparison that this series must also converge because we know any geometric series [tex]\sum\limits_n r^n[/tex] will converge as long as [tex]|r|<1[/tex].