We're asked to find the sum
of the given series,
which we will do using a known power series
for the functions listed below.
Notice the denominator is two n plus one factorial,
which is also the denominator for the power series
for sine x,
which means we will manipulate the given series
to fit the power series for sine x
to determine this infinite sum.
Notice the numerator is
x raised to the power of two n plus one,
and we have 16 raised to the power of n plus two.
We need the coefficient of this n to be two not one,
and therefore the first step will be
to write 16 as four squared.
The given series is equal to the sum for n equals zero
to infinity of negative one raised to the power of n
times four squared raised to the power of n plus two,
all over two n plus one factorial.
Notice now we have powers raised to powers,
which means now we can multiply the exponents.
Again, using the exponent rule,
a raised to the power of m raised to the power of n
equals a raised to the power of m times n.
So this series is equal to the sum
from n equals zero to infinity
of negative one raised to the power n.
And then here, when we multiply the exponents,
we distribute the two, giving us two n plus four.
So we have four raised to the power of two n plus four,
again, still all over two n plus one factorial.
Now looking back at the power series for sine x,
we want it to be in the form of x
raised to the power of two n plus one,
and we have four raised to the power of two n plus four.
Let's rewrite this exponent as two n plus one plus three,
which gives us the sum from n equals zero to infinity
of negative one raised to the power of n
times four raised to the power of two n plus one,
and then we still have plus three though
because the exponent is two n plus four.
But now we can rewrite four
raised to the power of two n plus one plus three
using the exponent rule a raised to the power of m times
a raised to the power of n equals a
raised to the power of m plus n.
Using this exponent rule, we can rewrite this
as four raised to the power of two n plus one times
four raised to the power of three over four cubed.
Let's go ahead and do that.
We have the sum from n equals zero to infinity.
We could have negative one raised to the power of n
and then times four
raised to the power of two n plus one times four cubed,
all over
two n plus one factorial.
Again, if we multiply here because the bases are the same,
we add the exponents, which does give us four
raised to the power of two n plus four.
Also notice in this form, four cubed is not affected by n,
which means we can now factor out four cubed
and write this as four cubed times the sum
from n equals zero to infinity
of negative one raised to the power of n
times four raised to the power of two n plus one,
all over two n plus one factorial.
Again, comparing this series to the power series for sine x,
we now have a perfect match where x is equal to four,
and therefore this series is equal to sine four,
giving us the product four cubed sine four
for the infinite sum of the given series.
And four cubed is equal to 64.
Let's write this as 64 sine four.
Let's also get a decimal approximation for this value.
Using the calculator,
we do want to make sure we are in Radian Mode
to evaluate sine four.
Let's press Mode.
The calculator's in Degree Mode right now.
We go down to the third row,
highlight Radian, press Enter.
And now go back to the home screen by pressing 2nd Quit,
and we enter 64 sine four,
which gives us approximately negative 48.4354.
While this is the decimal approximation,
it is always best to give the exact value when possible.
I hope you found this helpful.