Question

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

Answer 1

`\left|((-2)^n)/(3^n+1)\right|=(2^n)/(3^n+1)<(2^n)/(3^n)=\left(\frac23\right)^n`

As
`n\to\infty`, the sequence
`a_n=\left(\frac23\right)^n` converges to zero.

If you're talking about the infinite series


`\displaystyle\sum_(n\ge0)((-2)^n)/(3^n+1)`

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