Question

Determine whether the sequence converges or diverges. if it converges, find the limit. (if an answer does not exist, enter dne.) an = (5^n + 1)/ 9^n

Answer 1

`\displaystyle(5^n+1)/(9^n)=\left(\frac59\right)^n+\frac1{9^n}`

As
`n\to\infty`, both terms will approach 0, so the sequence converges. To show this, we can invoke the monotone convergence theorem.

Consider the function
`f(a,x)=a^x`, where
`0<a<1`. We have


`(\partial f(a,x))/(\partial x)=\ln a\,a^x=\ln a\,f(a,x)`

and since
`f(a,x)>0` for any such
`a`, and
`\ln a<0`, we have that
`(\partial f(a,x))/(\partial x)<0`, which means the function is decreasing over its entire domain.

To recap:
`f(x)>0` for all
`x`, so both
`f\left(\frac59,x\right)` and
`f\left(\frac19,x\right)` are bounded below. Then
`(\partial f(a,x))/(\partial x)<0` for all
`0<a<1`, which means
`f\left(\frac59,x\right)` and
`\f\left(\frac19,x\right)` are decreasing.

Therefore by the monotone convergence theorem, both sequences converge to 0, and so must
`a_n`.