Creating Maclaurin series

Finding the Maclaurin series for a function

Finding a series from standard results

Create a series for

We know the maclaurin series for :

Now we can just substitute in place of :

Next, we can expand + simplify the powers:

And then simplify the factorials and coefficient fractions:

Create a series for

• We know the maclaurin series for and :
• What if we substitute the series for into the series for ?
• Again, we can substitute the series for into each of these:

TODO: finish this stupid question, I can’t bear even looking at it anymore

Create a series for

• We know the maclaurin series for :
• We can rewrite as:
• Now we substitute in place of in the series for :
• Expand and simplify the powers:
• At the end, we combine this with to get the final series:

Create a series for the expression

• We know the maclaurin series for :
• We can substitute in place of in the series for :
• Expand and simplify the powers:
• Finally, we multiply the series by to get the final result: