Welcome to a video on the p-Series Test,
for convergence have an infinite series.
The p-series test, 
states that if we have an infinite series.
in the former of,
one over and to the power of P,
if p is greater than, one the series will converge
and if P is
 less than or equal to one, then the series diverges.
And, we alluded to this connection in the
integral test video,
where we looked when P was equal to one
versus,  P equal to one point one.
So, the p-series is a shortcut to apply an integral test,
if we have an infinite series that fits this form.
This test will be very helpful in the
next video, we talk about the comparison test.
So, we take a look at this infinite series,
it does fit the form for the p-series,
and since we can rewrite this as the summation,
from n equals one to infinity of one
over n to the power of one-half,
the exponent on n, which we call P, is equal to,
one-half which is,
less or equal to one.
Therefore, we know this infinite series will diverge.
So, we can state the infinite series diverges
by the p-Series Test.
So, it's a nice quick way to determine,
if a infinite series diverges or
converges, if it fits the correct form.
So, for this example, 
we can rewrite this as the summation
n equals four to infinity of,
one over this should be n to the,
remember its exponent over the index,
this should be four-thirds power,
so our powers equal to four-thirds,
which is greater than one
which is enough information to determine
that, this series will converge by the p-Series Test.
Again, this is a shortcut to the integral test.
if we let f(x) equal
 one over x to the four-thirds power,
we could apply the internal tests,
to show that it converges as well,
but this is obviously, much easier shorter way,
to show that it converges.
let's go and take a look at one more.
For this series, we can factor out the four,
then we have n the one-half in the numerator,
and n to the fourth in the denominator,
and if we going to use the p-series test ,
we cannot leave in this form
we wanna go ahead and rewrite this,
as four times the summation of
remember,
 if we're dividing here, we can subtract the exponents.
So, this should be,
n to the one-half minus four.
that would be 
n to the negative three and half
power
or negative seven-halves power,
which is the same as
one over and to the positive seven-halves power.
So, we have P equals seven-halves,
which is greater than one
so by the p-Series Test,
this series converges,
and multiplying it by four
does not affect whether, it'll converges or diverge.
So, the original series,
converges,
by the p-Series Test.
And, that's gonna do for this video.
and we will make use of this P-series
test again in the next video,
when we discussed the comparison test.
Thank you for watching.
