00:01
We want to make up an infinite series of non -zero terms who sum to 1, negative 3, and 0.
00:08
So let's just go ahead and start.
00:12
So in part a, what we're going to do is let's find it for something we do know, and then we can just multiply it by some constant c.
00:22
So what i'm going to do is multiply this by c, and i'm going to sum from n is equal to zero to infinity of just 1 over 2 to the n.
00:31
Now, over here on the side, i should probably remind you all that if we sum some geometric series like this, so n is equal to zero to infinity, and it's in the form of r to the end, this will sum to 1 over 1 minus r, as long as the absolute value of r is less than 1.
00:51
So in this case, our r is going to be 1⁄2.
00:54
So we can go ahead and write this as so of c times 1 over 1 minus 1⁄2, and one half or one minus one half is one half and then reciprocating that we get to c now over here we say we want this to be one so let's just set that equal to one dividing over by two we get c is equal to one half so for part a we can just go ahead and say that one half some or one half times n is equal to 0 to infinity of 1 over 2 to the n is equal to 1.
01:40
So we have part a here.
01:45
Now for part b, we're going to do something similar, but this time i'm going to choose to use a different one other than just one -half, because we could do the same thing we did before, get to the 2c, and then just set this equal to negative 3 as opposed to 2.
02:03
But since we want something to sum to zero, i'm going to make the work for us a little bit easier by just picking another one this time.
02:12
So this is going to be n is equal to zero to infinity, and this time i'm just going to choose one -third to the end.
02:21
Now, just like we did before, it's going to be c times 1 over 1 minus our r, and our r this time is going to be 1 -3rd.
02:32
So 1 -1 -1 -3 is 2 -thirds, and then reciprocating that we get 3 -5.
02:36
Half c all right now over here we want this to equal to negative three so we can multiply by two thirds and that would give us that c should just be negative two so if we come over here and let negative two bc and we sum this from n is equal to zero to infinity of one -third and this will give us negative three...