Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

60, -10, 5/3, -5/18, ...

Answer 1

I think it converges cause it keeps getting smaller

Answer 2

If we ignore the negative terms for the moment, can you see that every term is getting smaller and smaller by a factor of \(\dfrac{1}{6}\)?

Meaning,
\[60\times\frac{1}{6}=10\\
10\times\frac{1}{6}=\frac{10}{6}=\frac{5}{3}\\
\frac{5}{3}\times\frac{1}{6}=\frac{5}{18}\]
and so on.

Answer 3

yes

Answer 4

Okay, so if you keeping scaling a number down like this what value do you approach?

Answer 5

0?

Answer 6

Exactly right! If we accounted for the negative, we'd have a factor of \(-\dfrac{1}{6}\), but that won't change that conclusion.

What we've determined is that we can model this sequence with the formula
\[a_n=60\left(-\frac{1}{6}\right)^n\]
which is a geometric sequence. If you can determine that the ratio makes the next terms approach 0, then the sequence converges.

If you want a more formal approach to proving the limit exists, you can show the sequence is bounded and "monotonic", or that the sequence is Cauchy, but something tells me that's a bit beyond the scope of the question.

Answer 7

so it converges and the limit is 0

Answer 8

Yes

Answer 9

Awesome, thanks!!!! :D

Answer 10

yw!