Video Transcript
The function cos of 𝑥 can be
represented by the power series the sum from 𝑛 equals zero to ∞ of negative one
raised to the 𝑛th power divided by two 𝑛 factorial multiplied by 𝑥 to the power
of two 𝑛. Use the first two terms of this
series to find an approximate value of the cos of 0.5 to two decimal places.
The question tells us that the cos
of 𝑥 can be represented by the power series given to us in the question. What this tells us is that, for the
values of 𝑥 where the series converges, we have that the cos of 𝑥 is equal to the
sum from 𝑛 equals zero to ∞ of negative one to the 𝑛th power divided by two 𝑛
factorial multiplied by 𝑥 to the power of two 𝑛. What this means is we can attempt
to approximate the cos of 𝑥 by taking a partial sum. And we see that the question wants
us to do this for the first two terms of our series to find an approximate value of
the cos of 0.5 to two decimal places. So, we want our partial sum to only
be two terms.
Substituting in this information
gives us that the cos of 0.5 is approximately equal to the sum from 𝑛 equals zero
to one of negative one to the 𝑛th power divided by two 𝑛 factorial multiplied by
0.5 to the power of two 𝑛. Our sum starts at 𝑛 equals zero,
so the first term in our sum will be negative one to the zeroth power divided by two
multiplied by zero factorial all multiplied by 0.5 to the power of two multiplied by
zero. And our sum goes up to 𝑛 equals
one. So, the second term in our sum is
equal to negative one raised to the first power divided by two multiplied by one
factorial all multiplied by 0.5 to the power of two multiplied by one.
We’re now ready to start
evaluating. Any nonzero number raised to the
zeroth power is just equal to one. So, we have negative one raised to
the zeroth power is just one. We have that two multiplied by zero
is equal to zero, and zero factorial is just equal to one, so the denominator of our
first term is equal to one. And just as we had before, 0.5
raised to the zeroth power is just equal to one. Any number raised to the first
power is just equal to itself. So negative one raised to the first
power is just equal to negative one. Next, looking at the denominator in
our second term, we have that two multiplied by one is equal to two. And then two factorial is just
equal to two. So, we have a denominator of just
two.
Finally, we can see that 0.5 raised
to the second power is just equal to a quarter. This gives us one divided by one
multiplied by one plus negative one over two multiplied by a quarter. Which we can evaluate to be equal
to seven divided by eight, which, to two decimal places, is approximately 0.88. Therefore, we have shown by using
the first two terms of our power series the sum from 𝑛 equals zero to ∞ of negative
one raised to the 𝑛th power divided by two 𝑛 factorial multiplied by 𝑥 to the
power of two 𝑛 that we can approximate the value of the cos of 0.5 to two decimal
places as 0.88.
