How To Compute a Taylor Series Without a Calculator
The other day, I was working on a project which required the use of a Taylor Series. Unfortunately, the Taylor Series tends to be complicated and difficult to code. So I looked to brush up on my understanding of the concept.
During my pursuit to better comprehend the Taylor Series, I learned this method to compute the Taylor Series procedurally. Thus far, I have not seen any other blogs write about this method, so I am calling it “The Table Method”.
NOTE: A foundational explanation of the Taylor Series is out of scope for this blog. Here, we are only going to look at a method for evaluating a series. For more information, I recommend this video by 3blue1brown [https://www.youtube.com/watch?v=3d6DsjIBzJ4]
For this example, we are going to evaluate the function:
Here “a” is the value we evaluate, and f(x) is the function.
Pretty straightforward so far? Great!
Since we are looking at the Taylor Series of the above function, we could also write it like this:
Computers can evaluate the above function until the nth degree easily by leveraging the “for” loop, which is fantastic and impressive! But today we’re learning how to compute on paper; we must first understand the foundations and how to compute the Taylor series. Buckle up! Because we’re going to compute this problem by hand.
If you, dear reader, are a calculus student who struggles during tests trying to evaluate the series this method is very useful to you. Right when the test begins, do a “brain dump” and write schema (step 2) on your scratch paper. This will help guide you and prevent errors.
Now, let us get to the nuts and bolts of approaching this problem on paper.
Step 1) Write the question.
As a best practice, I always re-write the original question on my paper.
Step 2) Build the schema of your table.
Even though this is on paper, we can still create a ‘database’ that will store vital information to help us compute the Taylor Series.
NOTE: Notice that I moved the constant out and added it to the product column as c=34. This is a reminder for me to add the constant back in later.
Here is a breakdown of what each column does.
Index: this column tracks which order we are in.
f(x): store the current working function here. In the next iteration, the function will be the derivative of the first function.
I know that sounds really complicated, but here is a mathematical version of what I just said:
0) f(x)
1) f’(x)
2) f’’(x)
The function will become the derivative up to the nth degree of the index.
Plug: Here, we are going to plug “a” into the function as f(a). So for our naught index this will become sin(pi / 2).
X/n! : Spoken, you would say, “x over n factorial”. Here, x is the output from the plug column, and “n!” is the index as a factorial.
Term: x to the power of the index. You could also write it as such x^n
Product: The product is the output. In variable form, it could be written as constant * term * factorial.
Step 4) Write the naught index.
I also like to call this ‘establish the base’ because this is where our entire process starts from.
Step 5) derive the function, then loop through the table again.
Dearest reader, I hope that you are comfortable with derivatives because you shall be doing a lot of them with the Taylor Series. Look in column two and you’ll notice that each iteration is the derivative of the last function.
Besides that continue computing each column like before.
I stopped at the third power, but you could keep going forever or until you run out of coffee.
Here we learned how to compute the Taylor Series by hand using a method to build a database on your paper. When doing operations by hand, creating a table helps reduce errors by standardizing the process.
I hope this helps you better understand this subject.
