Converting explicit series terms to summation notation | Grade 11 | Math | Khan Academy (IN accent)

Added: 2026-06-02

[00:00:00]So, I have the series -5/3 + 25 over 6 - 125 over 9 plus and it just keeps going on and on and on forever. So, this right over here is an infinite sum or an infinite series.

[00:00:15]And what I want you to do right now is to pause this video and try to express this infinite series using sigma notation.

[00:00:24]So, I'm assuming you've given a go at it. Let's So, let's just look at each term of the series and let's see if we can express it with kind of an ever increasing index. So, the first thing that might jump out at you is this oscillating sign that's happening right over here. And whenever we see an oscillating sign, that's a pretty good idea that we could kind of view this as -1 to the nth where n is our index. So, for example, that right over there is -1 to the the first power.

[00:00:54]That is this right over here is -1 to the second power.

[00:00:58]That right over there is -1 to the third power. So, it looks like the sign is being defined by raising -1 to the index. Now, let's look at the other parts of these terms right over here.

[00:01:10]So, we have five.

[00:01:12]Then, we have 25.

[00:01:14]Then, we have 125. So, these are the powers of five.

[00:01:18]So, this right over here is 5 to the first power. This right over here is 5 to the second power. This right over here is 5 to the third power.

[00:01:25]So, this part, we're raising five to our index. Notice 1 1 2 2 3 3.

[00:01:31]And then finally, let's look at this.

[00:01:33]We have 3, 6, and 9.

[00:01:36]So, this literally if our index here is one, this is 3 * 1.

[00:01:41]If our index here is two, this is 3 * 2.

[00:01:45]If our index here is three, this is 3 * 3. So, this is 3 * 1. That is 3 * 2.

[00:01:51]Let me write it this way.

[00:01:54]3 * 2. That right over there is 3 * 3.

[00:01:57]So, this sets us up pretty well to write this in sigma notation. So, let's write it over here just so we can compare.

[00:02:04]So, let me give myself some real estate to work with.

[00:02:07]So, we could write this as the sum I'll do it in yellow.

[00:02:10]As the sum so this is our sigma. We can start our index n at 1 from n equals 1.

[00:02:17]We just keep going on and on forever.

[00:02:20]And so, it's -1 to the nth power.

[00:02:23]times 5 to the nth over Notice 5 to the nth over 3n is going to be equal to this. And you can verify when n equals 1 it's -1 to the nth power I'm sorry.

[00:02:38]-1 to the first power which is -1 * 5 to the first power which is 5 over 3 * 1.

[00:02:45]And we can do that for each successive term. And so, we're all done.