00:01
Given the 3 is on the form minus 1 power n and then x bar n plus 1 divided by square root and plus 3 goes from 0 to infinity.
00:11
In order to test for the convergence, we need to perform the limit ratio test here.
00:17
We'll have the limit n goes to infinity.
00:20
Am plus 1 divided by an.
00:25
And am plus 1 we should get equal to because this one is an absolute, so we don't need to care about the minus 1 and here, therefore we have the x, m dash 2 now over the square root the m plus 1 plus 3, dividing by an so we'll multiply by the reciprocal.
00:47
And now notice that we can cross out this one with the bow on the top here, and now we will have, this one is constant, coming outside, and we should get the absolute of x.
00:58
Inside we have the limit, angus to infinity, square root of m plus 3, divided by the square root of the n plus 1, 1 plus 3 and by the lovitan rule we will be easily to show that this limit here equal to 1 and therefore we get equal to absolute of x and you know to have a conversion we need to have this limit here to be smaller than 1 and then when this limit smaller than 1 implies that we have x is between minus 1 and 1 we need to test for the 2 endpoints here so when x equal to minus 1 we will have the series now becomes the minus 1 power n and minus 1 power n plus 1 over square root am plus 3.
01:46
So this one can be simplified and we get the minus 1 now will be better of the 2m plus 1.
01:53
2m plus 1 always odd and minus 1 bar 0 will be always minus so we can bring the minus outside and then 1 over square n plus 3.
02:02
So because this one will be the same rate of the pc of the 8 ,000 ,000 ,000 ,000 ,000 ,000 the b equal to 1 out of 2 smaller than 1 therefore this one will be divergent for the x equal to the 1 we get the series of the form minus 1 bower n and then 1 power n plus 1 1 divided by square and plus 3 and for this series here we have it will become here we have the it will add them up for the on the top here we have the same thing here will be the minus 1 power of the n because this one will equal to 1 here.
02:49
And then we have divided by square of n -3...