Original subtitles

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In this series we're talking about how to expand some through its first few terms.

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We've been given the series one over and squared so this value right here is our series And oftentimes

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we write our series as a 7 so we could say the series is a seven and a 7 is equal to 1 divided by end

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squared so that's our series The function that defines our series and this sigma notation here the thing

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that looks like any tells us take the sum of this series the value below the sigma notation is an equals

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1.

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This tells us start at the term that corresponds with N equals 1.

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This tells us where to stop so it tells us to stop at a value of and equals six.

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So if we expand this series through its first six terms we would take the values and equals 1 2 3 4

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5 and 6 the first six terms and add them together because we have this summation notation.

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Here we're taking the sum.

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So we add all those terms together that we find.

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So we're looking at the term that corresponds with end equals one and equals two three four five and

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six.

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Those are the first six terms.

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That's where the index tells us to start and stop.

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And all we have to do is start with N equals 1.

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Plug that into our series 1 divided by and squared and simplify that value.

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So first we'll say plug in equals 1.

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Here we're going to get 1 divided by 1 squared.

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So let's just go ahead and write that out for now.

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So 1 divided by 1 squared.

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Then we're going to plug in an equal to and because we're taking the sum because we have this summation

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notation we want to add all these terms together.

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So plugging in N equals 2 we're going to get 1 over 2 squared.

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Plugging in N equals three will get 1 over 3 squared and we'll get one over 4 squared one over five

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squared.

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And finally one over six squared and we stop there because this value right here of six tells us stop

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when you get two and equal sex.

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In other words your last term will be the term where you plug in and equal sex which is the term that

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we just found here.

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So that's all we're going to stop.

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So now we just need to simplify.

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So here in the denominator we get one square which is just 1 1 divided by one is 1.

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So here we get one then we're going to get plus two squared is force we get one fourth three squared

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is nine.

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So we get one ninth for score to 16 so one over 16 and then we're going to get one over 25 and one over

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thirty six and that's all there is to it.

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This value that we just found this series this sum is the exact same thing as what we were originally

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given.

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This notation right here it's just a different way of writing it.

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So this whole thing here is exactly equal to this whole thing here.

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This is just a simpler way of writing it.

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So if you were given this Sammet to start with and you wanted to write it in summation notation you

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could write it this way.

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And we'll talk about how to do that a little bit later.

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But here we just started with summation notation we were given this series and we were given the index

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and we expanded this series we expanded the summation notation into an expanded sum and that's how you

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expand a series through its first few terms.