Well, so the basic idea that we can have when we hear the words Maclaurin series calculator is that it is going to calculate some kind of series. However, the rest of it is not that easy. Let us first see what the Maclaurin series is. For that matter, let us begin with the Taylor series from where it originally comes.
So, in math, the Taylor series of a function gives an infinite sum of terms. Moreover, they find expression in terms of the function’s derivative at one single point. Therefore, in most cases, the sum of the function and its Taylor series is almost the same or very near to this point. Moreover, Brook Taylor is the founder of this series from where the name comes. He introduced the process back in 1715. However, it might be so that this single point in terms of which we are considering the derivatives is 0. Therefore, in that case, the Taylor series becomes the Maclaurin series. So, this series gets its name from Colin Maclaurin. He made a lot of use with this particular case of the Taylor series back in the mid- 1700s. So, the Maclaurin series calculator will be concerning this one.
Hence, the first n + 1 terms of a Taylor series give a partial sum. Therefore, this is a polynomial of the nth degree and we call it the nth Taylor polynomial of that function. Moreover, all Taylor polynomials are certain approximations of functions. Now, they get better with the increase in n. So, Taylor’s theorem helps in finding the quantitative estimates of the error that such approximations lead to. Anyway, we would not need any more definitions of the Taylor or Maclaurin series. So, what is more important is the application of a Maclaurin series calculator.
Maclaurin series calculator general form
However, before we get into the working of a Maclaurin series calculator, it is important that we understand the basic idea behind it. So, let us take a quick look at the general form of the series. Now, a Maclaurin series involves a function with some expansion series. Moreover, this yields the sum of derivatives of that particular function. Hence, the Maclaurin series of a function f(x) up to the order n can be found using the series [f, x, 0, n]. So, we already know that this is a special case of the Taylor series. In this case, x = 0. Therefore, we can write the general form of the Maclaurin series calculator in the form of-
f(x) = f (x) + f’ (x)(x – x) + + + …
So, we can combine the formula together and write-
f(x) =
So, here f(xo), f’(xo), f’‘(xo)… are all successive differentials when we have xo = 0.
Moreover, it might also be beneficial while using a Maclaurin series calculator if we know the Maclaurin series of some functions. So, let us take a quick look at some of them.
Functions
- For the exponent e^x, we have 1 + x + x^2 / 2! + x^3 / 3! + x^4 / 4! + …
- So next for sin x, we have (-1)^2 = x^(2k + 1) / (2k + 1)!
Therefore, this sums up to-
x – x^3 / 3! + x^5 / 5! + x^7 / 7! + …
- Well so after this we have cos x. Therefore, for this, the series will be ((-1)^n * x^2n) / (2n)! So, this basically gives- 1 – x^2 / 2! + x^4 / 4! – x^6 / 6! + …
- Now, we have 1 / (1 – x). So, for this one, the series will be x^k. So, this gives a simple series of 1 + x^2 + x^3 + x^4 + … However, here is a condition that we must follow. So, in this case, x must lie between -1 to + 1. So, it cannot be till infinity on either side. This is something we have to remember.
- Finally, we have a logarithmic function. So, let us take ln (1 + x). Therefore, for this one, the series will be (-1)^ (n + 1) * (x^n / n). So, the series looks like- x – x^2 / 2 + x^3 / 3 – x^4 / 4 + x^5 /5 – …
Maclaurin series calculator online
So, we have probably understood that handling the Maclaurin series calculator will not be a very easy job. This is because the concept of the Maclaurin series itself is not very simple. So, for this, you need to have a basic understanding of derivatives, factorials as well as arithmetic and geometric progress. Moreover, this is no easy job. That is a lot at once on the plate.
However, these days the internet makes everything easier. If you know how to use it properly, it can solve nearly all of your problems, and definitely problems regarding the Maclaurin series. So, you will easily find a Maclaurin series calculator online that you can use right in the beginning to understand the concept before you want to solve a sum. This is going to be very helpful because in math, unless and until you see the process, it does not become very clear in your brain. Therefore, let us check the working of a Maclaurin series calculator in detail for a better understanding.
Maclaurin series calculator Wolfram
So, most of us might already be familiar with the name Wolfram because of the wide selection of calculators it has. Moreover, the Maclaurin series calculator from Wolfram is one of the very few ones that are available online and are absolutely free. However, there is only one disadvantage. Wolfram has no built-in examples. So, if you want to learn the process right from scratch, the calculator cannot help you unless you have a question at hand. If you are willing to get a paid subscription, you can go for the Maclaurin series calculator from Symbolab. In addition to showing you all the steps, it will also give you examples even if you do not have a question.
Moreover, the beauty of online calculators is, for everything you have one, So, in the middle, if you cannot understand the factorial, say, you can have another calculator to help you out there, It is super easy and smooth.
Now, the Maclaurin series calculator is going to help you in many ways. It will help you learn the process. Moreover, you can solve a difficult sum. Finally, we are all human beings and manual work might have small errors that can prove to be significant for the sum. So, you can always use a Maclaurin series calculator to verify the solution. Moreover, this one shows you all the steps right from the beginning. Therefore, even if you have made a mistake, somewhere in the middle, you can easily find out if you cross-check all the steps. Now, let us take a quick look at the working of the Maclaurin series calculator that you can get from Wolfram with just a click.
Maclaurin series calculator with steps
The Maclaurin series calculator from Wolfram is very easy to use. Let us see how it works step-by-st
Step 1 to use a Maclaurin series calculator
First, we will go to the Wolfram website. Click on Wolfram. So, it will lead to a link that will land you directly on the page containing the calculator. Therefore, once you land on the page you will see a widget right in the middle of the page. There is no other elaborate toolbox. Hence, this is your Maclaurin series calculator. Moreover, you can customize the widget and update its color and other attributes. You can also embed it to prevent coming here again and again.
Step 2 to use a Maclaurin series calculator
So, inside the small rectangular widget that is the Maclaurin series calculator, there is a small box that looks like a search engine. Beside it, it is written, “Maclaurin series of”. So, you have to enter the function here whose series value you want to find out. Let us say the function is x^3 + 2x + 1. So, after you have entered, you will find an orange button right below it. It reads “submit”. Click it and the calculator will start computing.
Step 3 to use a Maclaurin series calculator
The Maclaurin series calculator will take a few seconds to compute the final value. Once the process is complete, you can see the entire section of the solution. So, at first, there is the interpretation of the input data. Then, it shows the expansion of the Taylor series at x = 0. Moreover, as it is a Taylor series, it converges everywhere. Therefore, in this case, it will be (x^3 + 1) + 2x + 0 (x^6). It will also find out the series expansion at whatever value of x you want it to. Moreover, you can also see more or less terms.
Now, the representation of the series will be 1 + 2x + x^3 will be ({ ) * x^n.
So, 1 + 2x + x^3 =1 (-1 + x)^n when n = 3, 3 (-1 + x)^n when n lies between 0 and 3 and 2(1 + x) (-1 + x)^n when n = 0.
So, this is the final answer that you can get from the Maclaurin series calculator. It is true that it might seem a little complicated but the more you use it, the easier it will become.
Maclaurin series to function calculator
So, we have already seen some of the standard functions and the Maclaurin series that they yield right at the beginning of the article. Moreover, you can do this with the help of a Maclaurin series calculator as well. Therefore, this will help you understand the process from where the formula comes. Moreover, it will also help you to solve complex cases of the standard functions that you might not be able to solve at once manually. So, go to the calculator at once!
Maclaurin series calculator FAQs
Are Taylor and Maclaurin series the same?
Ans. Well, it is like a Venn diagram. Every Maclaurin series is a Taylor series but the other way round is not true. So, the Taylor Series, or the Taylor Polynomial represents a particular function as an infinite sum of terms. Moreover, these terms can be calculated from the values of their derivatives at one single point. Now, the Maclaurin Polynomial is a single special case of the Taylor Polynomial. Therefore, here the single point to reference the derivative is 0. So, they cannot be exactly said to be the same or different either.
Who discovered the Maclaurin series?
Ans. Colin Maclaurin of the University of Glasgow discovered the Maclaurin series.
How do you calculate the Maclaurin series?
Ans. So, we already know that the Maclaurin Series is a particular Taylor series that is centered about 0. Moreover, the Taylor series can be centered around any number and is written as follows: ∑ n = 0 ∞ f ( n ) ( a ) ( x − a ) n n ! = f ( a ) + f ′ ( a ) ( x − a ) + f ′ ′ ( a ) 2 ! So, here the center number is a. If you want to calculate the Maclaurin series, the Maclaurin series calculator will consider it to be 0.
How do you find the interval of convergence of a Maclaurin series?
Ans. So, if you want to find the interval of convergence of the Maclaurin series, you have to erase the absolute value bars from the radius of convergence. However, before we can tell this to be the interval of convergence, it is important that we check if the series converges at one or both of the endpoints, that is, x = − 1 / 2 x=-1/2 x=−1/2 and x = 1 / 2 x=1/2 x=1/2.
What is the purpose of the Taylor and Maclaurin series?
Ans. So, a Taylor series is a concept that finds a lot of use in computer science, calculus, chemistry, physics, and most other areas that need higher-level mathematics. It is important because it can give a rough idea or an approximation or guess of what a function might end up looking like. Moreover, coming to the Maclaurin series is just a special case of the Taylor series. Therefore, the purpose remains pretty much the same. However, the reason why the Maclaurin is scooped out is that, with 0 as the center, there can be a lot of interesting cases.
