We want to find the partial sum.
Looking at our formula, notice how because we have
negative one raised to the power of n,
we have an alternating series
where when n is odd the term would be negative,
when n is even the term would be positive.
Also notice how the index n,
starts at n equals zero and goes to n equals 38.
Which means we have a total of 39 terms.
So because there are so many terms,
we don't want to find all 39 terms to find the sum.
We'll begin to expand the series
then hopefully recognize a pattern to make it easier
to find this partial sum.
By looking at our formula,
let's go ahead and factor out this three
to simplify the expansion of the series.
We can write this as three times the sum
from n equals zero to n equals 38
of negative one raised to the power of n times n.
We did not take advantage of this property
in the previous example,
but this time let's see how by factoring this three out
it simplifies the partial sum.
So this would be equal to three times the quantity
n equals zero,
we just have negative one to the zero times zero.
Plus when n is one we have negative one
to the first times one.
Plus when n is two, we have negative one
to the second times two.
When n is three, we have plus negative one to the third
times three.
Plus when n is four we have negative one to the fourth
times four.
And of course this series continues.
Let's go ahead and find the last two terms of the series.
N equals 37.
N equals 38.
So when n is 37 we have negative one to the 37th power
times 37.
Plus when n is 38 we have negative one to the 38th power
times 38.
Now let's go ahead and simplify.
So here we have three times the quantity,
this would be zero.
This is going to be negative one,
so we have plus negative one.
We could just write minus one, but I'll leave it as a sum.
Plus negative one to the second times two is positive two.
Plus negative one to the third times three
is negative three.
Plus negative one to the fourth times four is four.
And the last two terms, we have,
negative one to the 37th power which is negative one
times 37 which is negative 37.
Plus negative one to the 38th power is positive one
times 38 is positive 38.
Now to analyze this sum,
and see if we recognize a pattern.
Well the first term is zero, we're going to ignore that term.
Now if we pair the terms of the sum,
we'll recognize a pattern.
Notice how negative one plus two is one,
negative three plus four is also one.
Looking at the last pair, negative 37
plus 38 is also one.
So notice how this partial sum
is really just three times the sum
of a certain number of ones.
So if we can determine how many ones we're summing,
we can determine this partial sum.
Well we knew we had 39 terms,
but this first term is zero
which means we have 38 non-zero terms.
So if we pair 38 terms,
we'd have 19 pairs
because 38 divided by 2 equals 19.
Which means this partial sum is really
three times 19 ones or three times 19 times one,
which of course is just three times 19
which equals 57.
So the partial sum is equal to 57.
I hope you found this helpful.