[Course Instructor] We want
to find all the values of x
such that the given series would converge,
which is called the
interval of convergence.
Looking at our infinite series here,
notice how for this series,
this series has centered
at x equals seven,
because we have the
quantity x minus seven here.
We'll first apply the
ratio test given here,
where we know this limit
must be less than one,
for the series to converge.
This will give us an open
interval of convergence.
And then from there
we'll test the endpoints for convergence.
So before we apply the ratio test,
notice that a sub n is equal
to the quantity x minus seven
raised to the nth, divided
by a seven to the nth.
And then for a sub n plus one
would be equal to the
quantity x minus seven
to the power of n plus
one, divided by seven,
to the power of n plus one.
So now we'll apply the ratio test,
which would give us the limit,
as n approaches infinite
of the absolute value
of a sub n plus one, divided by a sub n,
instead of dividing here,
we'll multiply by the
reciprocal of a sub n instead.
So we first have a sub n plus one,
and then times the reciprocal of a sub n,
that'd be seven to the n,
divided by x minus seven to the n.
Now we'll simplify,
notice how here we have
one more factor of seven
in the denominator,
cause we have n plus one
factors of seven here,
and only n factors of seven here.
So this simplifies to one,
this simplifies to one factor of seven,
and notice here we have one
more factor of x minus seven
in the numerator,
so this simplifies to one,
this simplifies to the one
factor of x minus seven.
So now we have the limit,
as n approaches infinite
of the absolute value
of x minus seven, divided by seven.
We'll notice how this
is not affected by n,
as n approaches infinite,
and therefore this limit is
equal to the absolute value
of x minus seven, divided by seven.
In order for this series
to converge though,
this limit must be less than one.
So if we solve this absolute
value inequality for x,
this will give us the
open angle of convergence,
and then we'll test the endpoints.
So to solve this absolute
value inequality,
we can go ahead and
factor out positive 1/7,
and write this as 1/7 times
the absolute value of x
minus seven, that is less than one,
multiply both sides by seven,
and notice how here again, we can see,
this interval is going
to be centered at seven.
We can also see the
varities of convergence,
will be positive seven.
To solve this absolute value inequality,
this is telling us that x minus seven
must be less than seven,
and x minus seven must be
greater than negative seven.
So we add seven here.
We have x is less than 14,
and here we have x is greater than zero.
So the open interval of convergence
would be from zero to 14,
but the series my still
converge at the endpoints,
so now we'll test the endpoints.
So first, when x equals zero,
we would have the summation
from n equals one to infinite
of negative seven to the nth,
divided by seven to the nth,
and we can write this as
the summation from n equals
one to infinite of, we can
say, negative one to the nth,
times seven to the nth,
divided by seven to the nth,
if that's helpful.
Notice how this simplifies nicely,
and we have the summation
of n equals one to infinite
of negative one raised to the power of n.
We know this diverges for two reasons.
By the geometric series test,
the absolute value of r
would be equal to one,
and therefore the series diverges,
this also fails the
nth term diverges test,
so let's say by the geometric series test,
with the absolute value
of r equals, in this case,
the absolute value of negative one,
which is greater than or equal to one,
the series diverges at x equals zero.
And now we'll test that x
equals 14, when x equals 14,
we'd have the summation from
n equals one to infinite
of seven to the n,
divided by seven to the n,
no simplifying, because
the bases are the same,
we would subtract the exponents,
this would give us the
summation from n equals
one to infinite of seven, raised
to the power of n minus n,
that would be seven to the
zero, which equals one.
And by the nth term divergence test,
since the limit as
input as infinite of one
doesn't equal zero, the series
diverges at x equals 14.
So by the nth term divergence test,
the series diverges at x equals 14.
So that is how the series was
divergent at both endpoints,
and therefore the interval of convergence
remains the open interval from zero to 14.
So to answer the question,
the series is convergent
from x equals zero,
it does not include the
left endpoint, so we say no.
Two x equals 14,
it does not include the
right endpoint either,
so we say no again.
Again the interval of
convergence is the open interval
from zero to 14.
I hope you found this helpful.
