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Maclaurin Series Calculator

Power Series Calculator

Write down any function and the tool will take instants to compute its complex power series representation, with the steps shown.

Power series for:



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An online power series calculator is specifically programmed to produce the power series representation of a function (complex polynomial function) as an infinite sum of terms. You can convert a function to power series by using free power series expansion calculator.
For a better conceptual understanding, pay heed!

What Is a Power Series?

In the context of mathematical analysis,

“An infinite series that contains an infinite number of terms is termed power series expansion.”

Power Series Formula:

The following formula finds a power series representation for the function given.

$$ {\sum\limits_{n = 1}^\infty {{a_n}{x^n}} } = {{a_0} + {a_1}x }+{ {a_2}{x^2} + \ldots }+{ {a_n}{x^n} + \ldots } $$

The above formula is also taken into consideration by this best integral to power series calculator with steps to convert function to power series.

A series containing the factor \(\left(x – x_{0} \right)\) can also be considered for which the power series representation is given as follows:

$$ {\sum\limits_{n = 1}^\infty {{a_n}{{\left( {x – {x_0}} \right)}^n}} }= {{a_0} + {a_1}\left( {x – {x_0}} \right) }+{ {a_2}{\left( {x – {x_0}} \right)^2} +\ldots }+ {{a_n}{\left( {x – {x_0}} \right)^n} + \ldots ,} $$


‘a’= coefficient of the terms

‘n’= number of terms

Radius of Convergence(Power Series):

“It is the distance that is sketched from the centre of the convergent series to any end and can also be calculated by using this free radius of convergence power series calculator.”

Moreover, we have another radius of convergence calculator that is specifically designed to calculate this particular entity for any type of function series.

Ratio Test:

It is indeed the most effective way to find various parameters of power series which may include the following:

  • Interval of convergence
  • Radius of convergence
  • Interval of divergence

Generic Formula:

The basic equation that is applied to carry out the ratio test is as follows:

$$ L=\lim_{n \to \infty} \frac{a_{n+1}} {a_n} $$

The same formula is also used by our best power series from function calculator.

How to Analyse a Power Series?

Let’s resolve an example to analyse the power series.

Example # 01:

Determine the radius of convergence for the following power series function:

$$ \sum_{n=1}^\infty\frac{\left(x-6\right)^{n}}{n} $$


To determine the radius of convergence, we will apply the ratio test here as follows:

$$ C_{n}=\frac{\left(x-6\right)^{n}}{n} \hspace{0.25in} \text{Supposition} $$

Now if we take the value of the x as 6, the series will converge instantly. You can also verify it by using this best power series radius of convergence calculator. Anyhow,

let’s move forward!

$$ L= \lim_{n \to \infty}\frac{\left(x-6\right)^{n}}{n} $$

$$ L= \lim_{n \to \infty}[\frac{\left(x-6\right)^{n+1}}{n+1}* \frac{n}{\left(x-6\right)^n}] $$

$$ L= \lim_{n \to \infty}[\frac{\left(x-6\right)^{∞+1}}{∞+1}* \frac{∞}{\left(x-6\right)^∞}] $$

$$ L=\lim_{n \to \infty}[\frac{\left(x-6\right)^{1}}{1}* \frac{∞}{\left(x-6\right)}] $$

$$ \left|x-6\right| $$

The power series will converge for x-6 < 1

The power series will diverge for x-6 > 1

For this, the radius of convergence would be 1 that could be checked by subjecting to this p series calculator.

How a Power Series Calculator From Function Works?

With the help of our function to power series calculator, you get a proper expansion of the function for a desired number of variable x.

Let us see what you need to do:


  • First, enter a function in the menu bar
  • Select the type of the variable with which you wish to determine the power series
  • Set your entering point and the maximum order of the terms
  • Click ‘calculate’

The free function as power series calculator calculates:

  • A complete power series expansion of the given function


Do power series always converges?

As we know that the terms in the power series contain the variable x, the series may converge or diverge for particular values of x. For instance, if a power4 series is centred at x = a, it means that c0 gives the values of the series at x = a. That is why a power series always converges at its centre.

Does every function have a power series?

No, a function is only said to be a power series if it is infinitely differentiable. Here, you can also use our free power series finder to represent whether the function is differentiable or not.

What are power series good for?

You can find commonalities among various functions and also define new function series by using the power series. Rest this sum of power series calculator will also let you analyse any set of series in a brief manner.

Can we multiply power series?

Yes, you can multiply the power series of a function as it is just like the polynomial multiplication.

Is Taylor series a power series?

A Taylor series is always defined for a certain smooth function and cant be called a power series all the time. However, every power series is considered a Taylor series. Using our free online power series solution calculator can help you out in solving such series.

What is the centre of power series?

The centre of the power series is the value of the variable at which the series is centred. If you do not understand scenario, let this free center of power series calculator teach you properly with complete calculations shown.


Power series is used to represent and define common and new functions, respectively. Moreover, there is a vast use of the power series in the field of engineering science where it is used in numerical approximations of complex functions. However, a free online power series representation calculator with steps is an excellent approach for mathematicians to evaluate the sum of finite or infinite terms defined.


From the source of Wikipedia: Radius of convergence, Operations on power series, Analytic functions, Formal power series, Order of a power series.

From the source of khan academy: First order differential equations, Laplace transform, Second order linear equations.

From the source of lumen learning: Sequences, Indexing, Series, The Integral Test and Estimates of Sums, Comparison Tests, Alternating Series, Expressing Functions as Power Functions, Taylor and Maclaurin Series Summing an Infinite Series