This problem seem to bug many students, especially J1s who are doing Maclaurin’s series now. Many of the students are not sure how to find the coefficient of . They wonder how to “see” it. So here, I’ll attempt to show a direct method that doesn’t require us to stare and think.
Consider and we are interested to find the coefficient of .
First of all, to apply formula, we rewrite to . This step is based on indices and students should always check that the copied the correct power.
Now consider the formula above that is found in the MF15. We will take the coefficient of that’s found there, taking note that here.
We will end of with the following
Take note we must copy the alongside too and we substitute the in and preserve the .
Simplifying, we have
Here we attempt to factorise out, we need to figure how many (-1)’s we have here to factorise. To find out, we can take to find out how many terms there are. Next, we try to simplify the factorials.
Therefore, the coefficient is .
Note: cannot be further simplified as we do not know if is even or odd. In the event that is even, then and if is odd, then . An example is if we consider instead of , we observe that we will find a which is equals to 1 since .