00:03
So we're looking at the sum, infinite series, where n goes from 1 to infinity, n squared minus n plus 1 over n cubed minus 5.
00:17
And we're going to go ahead and use the comparison test.
00:23
Because if we just let this go, one thing we want to do is see, will the terms become, will the terms get smaller? in fact, go to 0 as well, and become infinite.
00:38
And one way to do that initially is to go just look at it, you know, in terms of the powers, like as n gets massive, this, you only need to look at the first term, the leading terms in each numerating denominator.
00:54
Because the plus one, the minus five, they don't do anything as any gets large.
00:57
The n is so small on comparison, n squared and n cubed.
00:59
So really, we're looking at n squared or n cubed, which reduces to 1 over n.
01:05
But this is the harmonic series.
01:06
And this, the sum of this, this sum of 1 over n as n goes from 1 to infinity, this diverges.
01:15
So it tells us this probably will diverge.
01:19
We'll do the comparison test just to make sure.
01:22
So the way we do that is we want to compare one term.
01:26
If you think about this in terms, right, we want to see if this term, this is term, i know, i plus 1, i plus 2, wherever you are in the series, i plus 3.
01:38
Oops should be i plus four whatever you want to take the term divided by the previous term so what that means we're going to do n plus one squared minus n plus one plus one divided by n plus one cubed minus five and we're going to divide that by the original expression n squared minus n plus 1 over n cubed minus 5 and then using just some fractions we end up with and i'm going to do some initial simplification here so we'll write the numerator out so it's n squared plus 2n minus or plus 1 minus n minus 1 plus 1 that's right here distribute the negative here and then that's over this which gives us n cubed plus 3 n squared plus 3 n plus 1 minus 5 so these are the this is the whole numerate and then we're going to multiply it by the reciprocal of this denominator so it's n to 3rd minus 5 over n squared minus n plus 1 and this we can simplify this a little bit but it's not really going to matter too much because we'll figure out what's going to happen here so n squared right let's see 2n minus and it's just going to be plus n plus 1, minus 1 plus 1.
03:18
So we do that plus 1.
03:20
Then we silver n cubed minus 5 all over this thing...