P series.
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When I say P series, this is what I mean.
Well I mean the series, sum n goes
from 1 to infinity of 1 over n to the p.
For some fixed real number p.
The question, then, is for which p does
the series converge.
The claim is the following.
This series converges
if p is digger than 1.
And it diverges, if p is less than or
equal to 1.
We can check this using Cauchy
Condensation.
Well lets just remember what Cauchy
Condensation says.
It says to analyze the convergence of a
given series,
it's enough to look at the convergence of
the condensed series.
In this case, I'm trying to analyse the
conversion to
this p series.
And here's the condensed series.
How did I get this?
Well, this is 2 to the n times the 2 to to
the nth term of the original series, okay.
So now, all I've got to do is
just determine whether this condensed
series converges or diverges.
How am I going to do that?
Well this series can be rewritten.
This series is the same as just 1 over 2
to the n to the p minus
1 power.
Look, I got 2 to the n in the numerator
and a power of 2 to the n in the
denominator.
So I can rewrite that power in the
denominator, as just the p minus 1 power.
Now this I could also rewrite.
I could rewrite this as the sum n goes 0
to infinity
of 1 over 2 to the p minus 1 to the nth
power.
Why is this an improvement?
What kind of series is this?
Right?
This series is just a geometric series
with common
ratio 1 over 2 to the p minus 1.
And by condensation, if this series
converges or
diverges, so too does the original p
series.
So I've reduced the entire question down
to just
knowing whether or not this geometric
series converges or diverges.
And I know how to analyse that.
If p
is bigger than 1, then this common ratio
is
less than 1 and in that case the series
converges.
And if p is less than or equal to 1,
then this common ratio is bigger than or
equal to 1.
And in that case the series diverges.
If you have already seen integrals, you
could
also do this by using the integral test.
So the integral test reduces the question
about this series is convergence,
to the question about this integral.
In this case, I'm trying to determine
whether or not
this series converges and that happens if
and only if the
integral from 1 to infinity of 1 over x to
the p, dx, has a finite value, if this
integral converges.
And this is true in this case, because
this function f of x equals 1
over x to the p, is positive and
decreasing on the interval that I care
about.
Okay, so all I've going to do now is
just analyse this integral and if this
integral has a finite value, this series
converges.
If this integral ends of being infinity,
this series diverges.
So let's see if I can compute this
integral.
So I'm trying to integrate from 1 to
infinity, 1 over x to the p, dx, and by
definition, this is the limit as n
approaches infinite
of the integral from 1 to big N of
1 over x to the p, dx.
And now, instead of writing 1 over x to
the p, I can write x to the minus p.
So this is the limit, n goes to infinity,
the integral
from 1 to big N of x to the minus p, dx.
Now I want to evaluate this.
So, I really want to assume that p isn't
1, because
I am going to write down an
anti-derivative of this and different
things happened right of p is, is 1.
But we've already handle the case were p
is 1 anyhow, that's just a harmonic
series.
Okay, so lets keep going assuming that p
is not 1, then I can evaluate this
definite
integral by finding an anti-derivative of
x to the
minus p, right, via the fundamental
theorem of calculus.
So this is the limit and goes to infinity,
what's an anti-derivative of x
to the minus p, x to the minus p plus 1,
over minus p plus
1, all right, then write it.
Add 1 to that exponent and I divide by
that same quantity.
And I'm evaluating this, from 1 to n.
So now I've just to plug in n and plug in
1 and take the difference.
So this is the limit begin, goes to
infinity of n to the minus p plus 1 over
minus p plus 1.
Minus what I get when I plug in a 1, which
is 1 over minus p plus 1.
I could combine these these two fractions,
this is the limit and goes to infinity
of n to the minus p plus 1 minus 1 over
minus p plus 1.
Okay, now the whole story is being told by
this limit.
This limit has a finite
value, then so does this integral and that
means that this series converges.
Well how do I check that?
Well the story is really being told by
this power here, right?
N to the minus p plus 1.
What happens?
So, if minus p plus 1 is positive, then n
to
a positive power, this is going to be very
large and well
then this limit then will be infinity.
That means that the series will diverge.
So then the series diverges.
If, on the other hand, minus p plus 1 is
negative and I'm taking a big number,
but I'm raising it to a negative power.
It's going to be ending up close to 0.
And in that case, this integral will have
a finite value and that means that
this series converges.
So then converges and maybe it's a little
bit
complicated to think about what these two
conditions are.
Instead of saying minus p plus 1 is
positive,
I could instead just say p is less than 1.
And instead of saying minus plus 1 is
negative,
I could just say that p is bigger than 1.
So now we've shown the whole
story right.
If p is less than 1 or equal to 1, the
series
diverges and if p is greater than 1, then
the P series converges.
Even when the P series converges, the
actual
values that we are getting can be quite
mysterious.
For example the sum of 1 over n squared
and goes 1 to infinity is pie squared over
6.
The sum of the reciprocals to the 4th
powers, the sum of 1 over
n to the 4th, n goes 1 to infinity is pie
to the 4th over 90.
The sum of the reciprocals of the 6th
powers, the sum the
goes from 1 to infinity of 1 over the n of
the 6th.
It's pie to the 6th over 945 so this is 2,
4, 6 we'll go to odd numbers,
all right, what's the sum of 1 over n
cubed, n goes from 1 to infinity.
Well, hundreds of years ago, Euler
calculated
an approximation.
He calculated some of the decimal digits
of this number.
And then not so long ago, Apery, in 1978,
showed that this was an irrational number.
But as far as we know, it's not a rational
multiple of pie cubed.
He gets stories pretty cool, right?
I mean, in the 1730s, Euler approximates
some number.
And over
200 years later, Apery comes along and
says
that number that you were approximating is
irrational.
I mean, to be doing mathematics means that
you're part of a community that will
outlive you.
You're joining into a conversation with
people from hundreds of years ago.
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