Video: Expressing a Series Using Sigma Notation

Express the series (1/2) + (1/4) + (1/6) + (1/8) + ... using sigma notation.

03:17

Video Transcript

Express the series one over two add one over four add one over six add one over eight and so on using sigma notation.

Let’s recall what sigma notation looks like. We use the Greek letter Ξ£. And next to it, we have the formula for the 𝑛th term, which I’ll call π‘Ž 𝑛. Underneath the Ξ£, we have the first value of 𝑛. So I’ll write 𝑛 equals π‘˜. But π‘˜ could be any number. It’s just the starting value of 𝑛. Above the Ξ£, we have the last value of 𝑛. Again, this could be anything; it could even be ∞.

So we’re going to express this series using sigma notation. So let’s start with writing Ξ£. We can see that the denominators here are going up in multiples of two. So our formula for the 𝑛th term is one over two 𝑛, starting with two times one. So underneath the Ξ£, we write 𝑛 equals one. And as the series has a dot dot dot, this implies we have infinite terms. So above the Ξ£, we write ∞ as that’s technically our last value of 𝑛. So we can express this series using sigma notation as the sum from 𝑛 equals one to ∞ of one over two 𝑛.

A useful tool which comes in handy when dealing with infinite series is index shifts. Recall that we call 𝑛 the index of summation. And we’ve seen that it doesn’t have to be the letter 𝑛; it could be a different letter. And we’ve seen that the starting value π‘˜ could actually be any number. You might often see it as 𝑛 equals zero or 𝑛 equals one, but actually it could be anything. For example, the sum from 𝑛 equals zero to ∞ of two to the 𝑛th power, the first term when 𝑛 equals zero will be two raised to the power of zero, which is just one. The second term, when 𝑛 equals one, will be two raised to the first power, which is just two. And this continues in that way.

But if instead we write the sum from 𝑛 equals one to ∞ of two to the 𝑛th power, we create a different series. The first term will be two raised to the first power because 𝑛 equals one is our starting value. So this will be two. And then it will continue on. But the important thing to note is that changing the first value of 𝑛 gives us a different series. And we can write the same series in a different way using index shifts.

For example, if we think about the series the sum from π‘˜ equals one to ∞ of π‘Žπ‘Ÿ raised to the power of π‘˜ minus one. Let’s rewrite this as a series which starts at π‘˜ equals zero. We can see that the first few terms of this sum are π‘Žπ‘Ÿ raised to the zero power add π‘Žπ‘Ÿ raised to the first power add π‘Žπ‘Ÿ raised to the second power and so on. So we could spot that actually we could rewrite this series as the sum of π‘Žπ‘Ÿ to the π‘˜th power starting at π‘˜ equals zero and going to ∞. So these are exactly the same series, but just written with a different starting value.

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