### Video Transcript

Expand the sum from π equals seven
to 11 of negative two plus seven over π.

This is the Greek letter β. And when we read this, we say the
sum from π equals seven to 11. So, given this information, how do
we find the sum from π equals seven to 11 of the expression negative two plus seven
over π? Well, weβre going to let π be
equal to seven. Weβll then let it be equal to
eight, nine, 10, and 11, and weβll find the sum of those expressions.

So when π is equal to seven, the
expression negative two plus seven over π is negative two plus seven over
seven. And thatβs negative two plus one,
which is simply negative one. So when π is seven, the expression
negative two plus seven over π is negative one. Letβs repeat this with π equals
eight. The expression becomes negative two
plus seven-eighths. To evaluate this, letβs write
negative two with a denominator of eight. Itβs negative sixteen eighths. Then we can add the numerators. And we find that when π is equal
to eight, our expression is negative nine-eighths.

Letβs now work out the value of
negative two plus seven over π when π is equal to nine. We get negative two plus
seven-ninths. Negative two with a denominator of
nine is negative 18 over nine. And then we see that the expression
is equivalent to negative eleven ninths. Next, we let π equal 10, and we
get negative two plus seven-tenths. We can then write that as negative
20 over 10 plus seven-tenths, which gives us negative thirteen tenths. In a similar way, when we let π
equal 11, we get negative 15 over 11.

And, of course, the question asks
us to expand our sum. So we simply need to add all of
these individual expressions. When we do, we get negative one
plus negative nine-eighths plus negative 11 over nine plus negative 13 over 10 plus
negative 15 over 11. And of course adding a negative is
the same as subtracting a positive, so we can rewrite this further.

When we expand our expression then,
we get negative one minus nine-eighths minus eleven ninths minus thirteen tenths
minus fifteen elevenths.