Video Transcript
Find the Maclaurin series of the
function 𝑓 of 𝑥 equals 𝑒 to the power of 𝑥.
We recall that the Maclaurin series
of a function 𝑓 of 𝑥 is given by the sum of the 𝑛th derivative of 𝑓 evaluated at
zero over 𝑛 factorial times 𝑥 to the 𝑛th power for values of 𝑛 between zero and
∞. So we’ll begin by differentiating
𝑓 of 𝑥 to evaluate 𝑓 prime of zero, 𝑓 double prime of zero, and so on. Ultimately, we’re hoping to find
what the 𝑛th derivative of our function is at 𝑥 equals zero. We know that 𝑓 of 𝑥 is equal to
𝑒 to the power of 𝑥. The derivative of 𝑒 to the power
of 𝑥 is simply 𝑒 to the power of 𝑥. So we find that 𝑓 prime of 𝑥 is
𝑒 to the power of 𝑥. 𝑓 double prime of 𝑥 is the
derivative of this, which is also 𝑒 to the power of 𝑥. And we can continue this. And we’ll see that the 𝑛th
derivative of 𝑓 will also be equal to 𝑒 to the power of 𝑥.
We need to work out what 𝑓 of
zero, 𝑓 prime of zero, and so on is. So 𝑓 of zero is 𝑒 to the power of
zero, which is one. 𝑓 prime of zero is also 𝑒 to the
power of zero, which is one. And since we know that the 𝑛th
derivative of 𝑓 at 𝑥 is 𝑒 to the power of 𝑥, we know that the 𝑛th derivative of
𝑓 at zero is 𝑒 to the power of zero, which is one. We therefore replace the 𝑛th
derivative of 𝑓 evaluated at zero with one. And we find that our summand is one
over 𝑛 factorial times 𝑥 to the 𝑛th power. We can, of course, simplify this to
𝑥 to the 𝑛th power over 𝑛 factorial. And so we see that the Maclaurin
series of the function 𝑓 of 𝑥 equals 𝑒 to the power of 𝑥 is the sum from 𝑛
equals zero to ∞ of 𝑥 to the 𝑛th power over 𝑛 factorial. And this is one plus 𝑥 plus 𝑥
squared over two factorial plus 𝑥 cubed over three factorial and so on.
So what about the radius of
convergence of this series? We recall that if a series
converges on the open interval from negative 𝑅 to 𝑅, its radius of convergence is
the number denoted by 𝑅. And of course, a power series
always converges absolutely within its radius of convergence. Similarly, the interval of
convergence is the open, closed, or semiclosed range of values of 𝑥, for which the
Maclaurin series converges to the value of the function.
