Summation Notation - Collapsing

Uploaded by TheIntegralCALC on 21.07.2011

Hi, everyone. Welcome back to Today we’re going to be talking about summation
notation. And in this particular problem, we’ve been asked to write this sum 1/2 +
1/4 + 1/8 + 1/16 + 1/32 + 1/64 in summation notation. So this I think takes a little bit
of practice and it gets a lot easier once you’ve done it a couple of times. But basically,
what we’re looking to do is collapse this series into summation notation so we’re
going to get this big summation notation here. We can pick any variable and in this case
I’ll pick n. But we’re going to have n equals something on the bottom. We also need
a number that’s going to appear up here at the top. And we need to write a function
here outside of the summation notation. And the easiest way to do this is to look
at your sum here and you’re looking for a pattern. So obviously, we can see them in
the numerator of all of these fractions. The numerator of each one is equal to 1 and we
can see that the denominator of each of these fractions is a multiple of 2. And if we look
at it more closely, we’ll probably start to see that we’ve got basically 2 to the
first power, 2 to the second power, 2 to the third power etc. because 2 to the first is
2, 2 squared is 4, 2 cubed is 8 and on down the line. And so we’re looking for that
kind of a pattern and we can see it here if we look at this pattern that we’ve just
written out. We’ve got 1, 1 and 1 over 2 to the 1st, 2 to the 2nd, 2 to the 3rd. So
we know that our function is going to be 1/2 to the n when n is equal to 1. So that means
the thing that’s changing is this exponent here. The numerator is always going to stay
the same. The base in the denominator always stays the same. The only thing that’s changing,
that’s counting up is this exponent right here. So what we do is we make this series
here, this 1, 2, 3, that becomes n, the variable. And then we need to know what the first number
in that series is and clearly here, we can see it’s 1. So we set n =1 in the bottom
here. That means that n starts at 1. Now what we need to know is how far n goes
up to. So 2 to the 6th power is 64 which means n is going to go all the way up to 6. So that
is our summation notation. If we just clean this up, we get n = 1, n starts at 1, it goes
up to 6 and we’re going to be summing up 1/(2^n). So that’s it. this is our final
answer. That’s the series or the sum we’ve written in summation notation. So again take
some practice to kind of recognize those patterns. That’s what you’re looking for. It’s
the things that stay constant and then the parts of the series that are going to be counting
up or counting down depending on your series. And that’ll determine what you pick for
your variable and what these numbers here end up being. So that’s it. I hope this
video helped you guys and I will see you in the next one. Bye!