Worked example: divergent geometric series | Series | AP Calculus BC | Khan Academy

So we've got this infinite series here, and let's see. It looks like a geometric series. When you go from this first term to the second term, we are multiplying by -3, and then to go to the next term, we're going to multiply by -3 again.

So it looks like we have a common ratio of -3. We could actually rewrite this series as being equal to 0.5. I could say times -3 to the 0 power, -3 to the 0 power plus plus 0, or maybe I could just keep writing this way: - 0.5 * -3 to the 1 power, * -3 to the 1 power - 0.5, - 0.5 * -3 to the 2 power, -3 to the 2 power.

And we're just going to keep going like that. We could just say we're just going to keep having -0.5 * -3 to each or to higher and higher and higher powers, or we could write this in Sigma notation. This is equal to the same thing as the sum from, let's say, n equals 0 to Infinity.

It's going to keep going on and on forever. And it's going to be this first, it's going to be, you could kind of think the thing we're multiplying by 3 to some power. So it's going to be -0.5. Actually, let me just do that yellow color, so it's going to be 0.5 times -3. Negative? Let do that blue color, so times -3 to the nth power.

Here this is when n is zero, here is n is one, here is n is equal to two. So we've been able to rewrite this in different ways, but let's actually see if we can evaluate this.

So we have a common ratio of -3. Our R here is 3. The first thing that you should think about is, well, in order for this to converge, our common ratio, the magnitude of the common ratio, or the absolute value of the common ratio, needs to be less than one for convergence.

And what is the absolute value of -3? Well, the absolute value of -3 is equal to 3, which is definitely not less than one. So this thing will not converge. This thing will not converge.

Even if you look at this, it makes sense because the magnitudes of each of these terms are getting larger and larger and larger. We're flipping between adding and subtracting, but we're adding and subtracting larger and larger and larger and larger values.

Intuitively, when things converge, you're kind of, each successive term tends to get diminishingly small, or maybe it cancels out in some type of an interesting way. But because the absolute value of the common ratio is greater than or equal to one in this situation, this is not going to converge to a value.