yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


2m read
·Nov 11, 2024

Let's get some practice taking sums of infinite geometric series.

So, we have one over here, and just to make sure that we're dealing with the geometric series, let's make sure we have a common ratio.

So, let's see: to go from the first term to the second term, we multiply by ( \frac{1}{3} ). Then, to go to the next term, we are going to multiply by ( \frac{1}{3} ) again, and we're going to keep doing that.

So, we can rewrite the series as ( 8 + 8 \times \frac{1}{3} + 8 \times \left(\frac{1}{3}\right)^2 + 8 \times \left(\frac{1}{3}\right)^3 + \ldots ). Each successive term we multiply by ( \frac{1}{3} ) again.

So, when you look at it this way, you're like, okay, we could write this in sigma notation. This is going to be equal to…

So, the first thing we wrote is equal to this, which is equal to the sum:

The sum can start at zero or at one, depending on how we'd like to do it.

We could say from ( k = 0 ) to infinity. This is an infinite series right here; we’re just going to keep on going forever. So, we have:

[
\sum_{k=0}^{\infty} 8 \times \left(\frac{1}{3}\right)^k
]

Let me just verify that this indeed works, and I always do this just as a reality check, and I encourage you to do the same.

So, when ( k = 0 ), that should be the first term right over here. You get ( 8 \times \left(\frac{1}{3}\right)^0 ), which is indeed ( 8 ).

When ( k = 1 ), that's going to be our second term here. That's going to be ( 8 \times \left(\frac{1}{3}\right)^1 ), which is what we have here.

And so, when ( k = 2 ), that is this term right over here. So, these are all describing the same thing.

Now that we've seen that we can write a geometric series in multiple ways, let's find the sum.

Well, we've seen before, and we proved it in other videos, if you have a sum from ( k = 0 ) to infinity and you have your first term ( a ) times ( r^k ), assuming this converges—so, assuming that the absolute value of your common ratio is less than one—this is what needs to be true for convergence.

This is going to be equal to:

[
\frac{a}{1 - r}
]

This is going to be equal to our first term, which is ( a ), over ( 1 - r ).

If this looks unfamiliar to you, I encourage you to watch the video where we derive the formula for the sum of an infinite geometric series.

But just applying that over here, we are going to get:

This is going to be equal to ( \frac{8}{1 - \frac{1}{3}} ).

We know this is going to converge because the absolute value of ( \frac{1}{3} ) is indeed less than one.

So this is all going to converge to:

[
\frac{8}{1 - \frac{1}{3}} = \frac{8}{\frac{2}{3}} = 8 \times \frac{3}{2} = 12
]

Let's see: this could become, divide ( 8 ) by ( 2 ); that becomes ( 4 ), and so this will become ( 12 ).

More Articles

View All
Forget Scarecrows—Falcons Protect This Farm | National Geographic
We’re kind of like security guards. We arrived before the sugar content of the fruit starts going up. As the foods ripen, the birds are more and more attracted to it, so we stand guard ten hours a day in that field until basically the fruit is harvested. …
How Do Chameleons Change Color?
There is a misconception about chameleons that they change their color in order to blend in with their environment. That is actually not the case. When a chameleon is calm, it is green, and so it naturally blends in with its leafy surroundings. But male c…
Derivatives of sec(x) and csc(x) | Derivative rules | AP Calculus AB | Khan Academy
In a previous video, we used the quotient rule in order to find the derivatives of tangent of X and cotangent of X. What I want to do in this video is to keep going and find the derivatives of secant of X and cosecant of X. So, let’s start with secant of …
Turning Gravity Into Light - Smarter Every Day 146
Hey, it’s me Destin. Welcome back to Smarter Every Day. If you’ve ever watched Smarter Every Day, you know that I spend a lot of time off the grid. Right now I’m in the Amazon rainforest, and I don’t really know what this thing is. I think it’s some kind …
8 steps to get your sh** together
Here are eight steps to get your together. Step one: Tell no one. The urge to tell everybody you know, especially the people closest to you, about this big change you’re going to make is often overwhelming. Because it feels really good to announce to eve…
Food and energy in organisms | Middle school biology | Khan Academy
Hey, quick question for you. You ever look at a person’s baby pictures and wonder how people go from being small to, well, big? I mean, yes, I get it; people grow up, but here I’m thinking more on the level of the atoms and molecules that make up the body…