Question

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

60, -10, five divided by three, negative five divided by eighteen, ...
Diverges

Converges; 11100

Converges; 72

Converges; 0

Answer 1

Try to find a closed form expression for the \(n\)-th term in the sequence. Hint: it's a geometric sequence, find the common ratio.

Answer 2

@SithsAndGiggles i dont understand how to even start it

Answer 3

First term = 60

Divide by -6 (i.e. multiply by -1/6) and you get the second term = -10.

Divide again by -6 (or multiply by -1/6) and you get the third term = (-10)/(-6) = 5/3.

Thus the common ratio is \(-\dfrac{1}{6}\). There's a certain trait about common ratios that satisfy \(|r|<1\) that will help you figure out the rest.

Answer 4

Diverges

Converges; 11100

Converges; 72

Converges; 0
so how do i determine if one of these is the answer ? @SithsAndGiggles

Answer 5

If the ratio satisfies \(|r|<1\), then the sequence converges. Does \(-\dfrac{1}{6}\) satisfy this?

Answer 6

im not sure

Answer 7

What is \(\left|-\dfrac{1}{6}\right|\)?

Answer 8

1/6

Answer 9

Right, and 1/6 is less than 1. So this sequence converges.

Answer 10

converges at what ?

Answer 11

Think about what's happening to the each next number in the sequence. You're making the value smaller and smaller.

As an example, what happens if you were to cut a pie in half and ate one of the halves, then again cut in half, and again and again and so on. What would you be left with at the end of all this cutting?

Answer 12

This scenario basically illustrates the geometric sequence \(\left\{1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{6},\cdots\right\}\). The terms get smaller and smaller, but never become negative. This sequence converges to 0.