Enter your function and the point of expansion to find the power series expansion of your function up to a specified order (n).
This power series calculator allows you to expand a function into a power series for a given variable and also shows you the step-by-step calculations involved. With it, you can:
This calculator is very helpful for analyzing and approximating the function values.
Limitation: The calculator can handle a range of mathematical functions, but this might not work for functions with discontinuities or infinite complexity.
"A power series is an infinite series where every single term is formed by the multiplication of a constant coefficient (cn) and a variable (x) raised to an increasing non-negative integer power (n), often centered around a value 'a'.”
∑n=0∞ cn(x-a)n = c0 + c1(x-a) + c2(x-a)2 + c3(x-a)3 + .......;
Where:
When the value “a” is zero, then the series becomes simpler as:
∑n=0∞ cnxn = c0 + c1x + c2x2 + c3x3 + ......;
This series proceeds to represent a function within the interval of convergence, and it diverges outside the interval. They are used:
A power series centered at a point converges for a value of x within a certain interval. Within this interval of convergence, the absolute value of the terms typically decreases as the power increases. Hence, their sum approaches a finite value. This power series convergence can be found by using the ratio test.
∑(xn) = 1 + x + x2 + x3 + ... ∑n=0∞ xⁿ
The power series converges when the absolute value |x| < 1.
At this point, its value becomes 1/(1-x). We can express the function as power series manually, as we have done below. To quickly obtain its power series representation, try our power series calculator, which takes a function as input and outputs its equivalent power series.
f(x) = 1/(1 - x) = Σ(n=0 to ∞) xn = 1 + x + x2 + x3 + ..., |x| < 1
This equation shows that the series converges at a certain value, and we can get another function by replacing x with -x
f(x) = 1 / (1 + x) = ∑n=0∞ (−x)ⁿ = 1 − x + x² − x³ + … ∣−x∣<1⟹∣x∣<1
Now, let's consider the case where x=0.3. Since ∣0.3∣<1, we know that the geometric series ∑n=0∞(0.3)n converges. To find the finite value it converges to, we use the formula for the sum of a geometric series:
∑(xⁿ) = 1+0.3+(0.3)²+(0.3)³+⋯=1+0.3+0.09+0.027+…
The sum of this series is:
1/1−0.3 = 1/0.7 =10/7
Using x like 0.3 in the general power series gives us the numerical value that the series converges to at that point.
For advanced power series analysis, including expansions of complex functions up to a specified order, use our power series representation calculator.
To find power series representation for the function, write the function as an infinite series containing a variable raised to a whole-number exponent. Follow these steps to expand the series manually:
Let's find the power series expansion for f(x)=eˣ
Solution:
The key to finding the power series representation of a function is its derivatives. So, evaluate the function's derivatives at x = 0.
Step # 1: Write Out the General Form
f(x) = f(a) + (f′(a)(x − a)) / 1! + (f″(a)(x − a)²) / 2! + … + (fⁿ(a)(x − a)ⁿ) / n!
Step # 2: Determine the Coefficients
We find the derivatives of f(x) that is evaluated at x = 0
f(x) = eˣ
f'(x) = eˣ derivative of (eˣ) is itself
f”(x) = eˣ derivative of f'(x) = derivative of eˣ
fⁿˣ = eˣ) nth derivative of (eˣ) is always (eˣ)
Therefore, the coefficients f(0), f'(0), f''(0), ... , f^(n)(0) will all be 1.
Step # 3: Substitute Coefficients into the Series
The power series of function f(x) is given by:
f(x) = f(a) + (f′(a)(x − a)) / 1! + (f″(a)(x − a)²) / 2! + ... + (fⁿ(a)(x − a)ⁿ) / n!
In this case,
Step # 4: Expand the Series
Put the values into the general form to convert function to power series:
f(x) = 1 + x / 1! + x² / 2! + x³ / 3! + … + xⁿ / n!
Step # 5: Write Out the Expanded Series
The series can be written more concisely using summation notation:
f(x) = ∑n=0∞ xⁿ / n! from n = 0 to ∞
This represents the sum of all the terms from n = 0 to infinity, where each term is raised to the power of “n” divided by its factorial (n!). Therefore, the power series expansion from n = 0 to ∞ is:
∑ xⁿ / n!
Determining the coefficient can involve various methods that are based on the function. However, it can be easily calculated with a power series coefficient calculator.
Here’s how our power series calculator works to create the power series from function:
Power series has significant applications over various fields, including:
As we know the power series has a variable x in which the series may converge for a certain x value and diverge for others. When x equals a, the power series centered at x=a is represented by c0. It is evident in the terms that simplify to zero. Therefore a power series has convergence at its center.
No, not at all every function has a power series representation. These are the reasons why a function might not have a power series:
Every Taylor series is a power series, but not every power series is a Taylor series. A Taylor series is always defined for a certain smooth function.
References:
From the source of Wikipedia.org: Power Series.
From the source of Math.Libretexts.org: Power Series and Functions.
Links
Home Conversion Calculator About Calculator Online Blog Hire Us Knowledge Base Sitemap Sitemap TwoSupport
Calculator Online Team Privacy Policy Terms of Service Content Disclaimer Advertise TestimonialsEmail us at
[email protected]© Copyrights 2025 by Calculator-Online.net