# Question Video: Expressing a Series Using Sigma Notation Mathematics • Higher Education

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.

Letβs recall what sigma notation looks like. We use the Greek letter Ξ£. And next to it, we have the formula for the πth term, which Iβll call π π. Underneath the Ξ£, we have the first value of π. So Iβll write π equals π. But π could be any number. Itβs just the starting value of π. Above the Ξ£, we have the last value of π. Again, this could be anything; it could even be β.

So weβre going to express this series using sigma notation. So letβs start with writing Ξ£. We can see that the denominators here are going up in multiples of two. So our formula for the πth term is one over two π, starting with two times one. So underneath the Ξ£, we write π equals one. And as the series has a dot dot dot, this implies we have infinite terms. So above the Ξ£, we write β as thatβs technically our last value of π. So we can express this series using sigma notation as the sum from π equals one to β of one over two π.

A useful tool which comes in handy when dealing with infinite series is index shifts. Recall that we call π the index of summation. And weβve seen that it doesnβt have to be the letter π; it could be a different letter. And weβve seen that the starting value π could actually be any number. You might often see it as π equals zero or π equals one, but actually it could be anything. For example, the sum from π equals zero to β of two to the πth power, the first term when π equals zero will be two raised to the power of zero, which is just one. The second term, when π 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 π equals one to β of two to the πth power, we create a different series. The first term will be two raised to the first power because π 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 π 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 β of ππ raised to the power of π minus one. Letβ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π raised to the zero power add ππ raised to the first power add ππ raised to the second power and so on. So we could spot that actually we could rewrite this series as the sum of ππ to the πth power starting at π equals zero and going to β. So these are exactly the same series, but just written with a different starting value.

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