### Video Transcript

Which of the following series is not
equivalent to the sum from π equals one to β of π π? π one add π two add the sum from π
equals zero to β of π π add three. π one add the sum from π equals two
to β of π π. The sum from π equals zero to β of π
π add one. π one add π two add the sum from π
equals one to β of π π add three. Or π one add π two add the sum from
π equals three to β of π π.

Letβs start by having a think about
what this series looks like. The first term of the series, the sum
from π equals one to β of π π, will be π one because our starting value is π equals
one. And our second term will be π two and
our third term will be π three. And it will continue in this way. So letβs have a look at each of the
five series that weβve been given. For this first series, we have π one
add π two. And then weβre adding the series the
sum from π equals zero to β of π π add three. So the starting value of π is
zero. So the first term of the series will be
π zero add three, which is just π three.

The next term will be when π equals
one. So this will be π one add three. So this will be π four. And as this goes up to β, this will
just continue in this way. For the second series, this starts with
π one and then we have the series the sum from π equals two to β of π π. So the first value will be when π
equals two. So this will be π two. And then the next value will be π
three and again this goes up to β. So this will continue in this way.

The third series is the sum from π
equals zero to β of π π add one. So the first term will be when π
equals zero. So weβll have π zero add one, which is
just π one. Then when π equals one, weβll have π
two. And this will continue as the last
value of π is β. If we then have a look at the fourth
series here, we have π one add π two. And then we have the series the sum
from π equals one to β of π π add three. So this series starts when π equals
one. And if we substitute that into the
formula for the πth term, we have π one add three, which gives us π four. And then when π equals two, weβll have
π two add three, which is π five. And the last value of π is β. So this will continue in this way.

So as weβve missed out π three, it
looks like this one is not equivalent. But letβs double check the last series
to be sure. This series starts with π one add π
two and then we have the series the sum from π equals three to β of π π. So when π equals three, we have π
three; when π equals four, we have π four; and so on. So we can say that the series which is
not equivalent to the sum from π equals one to β of π π is π one add π two add the
sum from π equals one to β of π π add three.

So weβve seen some infinite series
which sum to β and some which sum to a finite number. And in fact, thereβs two really
important words in mathematics which we use to describe series, convergent and
divergent. A convergent series is one where the
value of the series is a finite number, such as the series we saw earlier which gave us
the value of one. A divergent series is one where we say
the value of the series is β. It could be the positive or negative β,
such as the series the sum from π equals one to β of π.