Power Series Calculator

An online power series calculator is specifically programmed to produce the power series representation of a 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 find a power series representation for the function given.

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

A series containing the factor \(left( {x – {x_0}} \right\) can also be considered for which the power series representation is given as follows:

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

where;

‘a’= coefficient of the terms
‘n’= number of terms

How a Power Series Calculator 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:

Input:

• 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’

Output:

The free power series calculator calculates:

• A complete power series expansion of the given function

FAQ’s:

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 centered 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 center.

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.

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.

Conclusion:

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 is an excellent approach for mathematicians to evaluate the sum of finite or infinite terms defined.

References:

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