00:02
Okay, let's take a look at this infinite series.
00:06
We are going from zero to infinity, and we have minus 1 to the n, 2x plus 8 to the end.
00:15
And we're going to look at finding radius, interval of convergence, converging, absolutely converging, conditionally, those kind of things.
00:24
So to do all that, we are going to start with doing the ratio test.
00:29
All right, so the ratio test basically says if i take the limit as n goes to infinity of the absolute value of the next term over the current term, the nth term, then if i can find values of x for less than one, then our series converges.
00:48
So we are going to do that.
00:50
All right, so we are doing absolute value so that minus 1 to the n term, when you do absolute values, the same as just multiplying by 1.
00:59
So i'm not going to write that in there, but the rest of it needs to go in.
01:04
This is the n plus 1 term over the current term.
01:11
Okay, so now we can do exponential rules.
01:16
I have the same base, so i can subtract the exponents, and i end up with 2x plus 8 to the first power on top.
01:24
And now that we don't have any ends left, we can simply rewrite it as without the limit sign.
01:31
Absolute value of 2x plus 8 and then we set this less than 1 for convergence.
01:38
Okay, so let's keep working it.
01:39
I can break that up as minus 1, less than 2x plus 8, less than 1.
01:45
Then i can subtract 8 from both sides.
01:49
That'll give me minus 9, less than 2x, actually subtract 8 from all three parts.
01:55
Okay, so i get minus 9, 2x, and on the right side i get minus 7.
01:59
Finally, i can divide by two, so i get minus nine halves, less than x, less than minus seven halves.
02:08
All right, so with the ratio test, we do have the check end points to find our full interval of convergence.
02:15
Though i can tell you right away, the radius convergence is half the distance in between minus nine halves and minus seven halves, and that a half distance is simply one half.
02:27
Okay, so we have the radius of convergence, but we still need to find our interval, and to do that, we need to check endpoints.
02:36
Let's write this down, check end points.
02:42
Okay, so let's go ahead and first check the left end point, and we plug it into our series to see what we get.
02:50
We have minus one to the end.
02:53
If i plug in minus 9 has for x, i get this form.
02:59
And we can clean that up.
03:02
This will be equal to minus 1 to the n.
03:07
And inside i get minus 9 plus 8 or minus 1 to the end, which just gives us 1 to the n or just the sum as n goes to 0 of, and equals 0 to infinity of just 1 to the n, which is just 1.
03:24
And this diverges...