Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)

an = (−3^n)/(4n!)

Answer 1

`a_(i) = ((-3)^(i))/(4\cdot i!)` converges.

Explanation:

The convergence analysis of this sequence is done by Ratio Test. That is to say:

`r = (a_(n+1))/(a_(n))`, where sequence converges if and only if
`|r| < 1`.

Let be
`a_(i) = ((-3)^(i))/(4\cdot i!)`, the ratio for the expression is:

`r =-(3)/(n+1)`

`|r| = (3)/(n+1)`

Inasmuch
`n` becomes bigger, then
`r \longrightarrow 0`. Hence,
`a_(i) = ((-3)^(i))/(4\cdot i!)` converges.