Power Series and Convergence

Many functions encountered in calculus and complex analysis can be expressed as power series of the form ${\sum }_n^{0\infty }{a}_n{z}^n$. When such a series converges in some disc centered at the origin, we speak of its radius of convergence $R$. Inside that disc, the series defines an analytic function. Beyond it, the series may diverge or converge conditionally, and understanding where the series breaks down is crucial for both theoretical insights and practical computations.

Two standard tests offer ways to determine $R$. The root test states that $R=\frac{1}{\underset{n\to \infty }{\lim \sup }{\left|a_n\right|}^{\frac{1}{n}}}$, where $\lim \sup$ denotes the limit superior. Alternatively, the ratio test often proves simpler when the coefficients satisfy a convenient recurrence: $R=\underset{n\to \infty }{lim}{\left(\frac{\left|a_{n+1}\right|}{\left|a_n\right|}\right)}^{-1}$. For many common series, one of these limits can be evaluated exactly, yielding an explicit radius. In more complicated situations we may approximate $R$ numerically using a finite sequence of coefficients.

Approximation Strategy

This calculator estimates the radius of convergence using a truncated version of the root test. Suppose we have coefficients up to $a_N$. The calculation involves computing ${|}^{a_k}\frac{1}{k}$ for each $k$ in that range and taking the maximum value to approximate the $\lim \sup$. The reciprocal of this maximal root provides an estimate for $R$. Though approximate, this approach converges as we supply more coefficients. It is especially useful for power series defined by complicated recursions or by numerical computation, where a symbolic solution would be difficult.

Because the radius of convergence is controlled by the growth rate of coefficients, extreme values or irregular patterns may distort the estimate if we only consider a few terms. Nonetheless, this tool offers quick intuition. If the sequence of coefficients stabilizes to a recognizable pattern, the approximation becomes increasingly reliable. In teaching contexts, computing $R$ numerically can help students visualize the disc of convergence and see how it connects to analytic continuation and singularities of the underlying function.

Using the Calculator

Enter coefficients separated by commas, such as 1, -1/2, 1/3, -1/4. Fractions and decimals are allowed. The program parses each term with math.js, evaluates its absolute value, and calculates the $k$-th root. The resulting estimates determine the radius according to the algorithm above. The final number reflects how far from the origin you may reliably evaluate the series before divergence occurs.

As you experiment, observe how doubling or halving coefficients affects $R$. The interplay between coefficient growth and convergence radius mirrors the link between local behavior of analytic functions and the location of singularities in the complex plane. For instance, the Taylor series of $1/(1-z)$ has coefficients that do not decay and thus possesses radius $1$. More rapidly growing coefficients produce smaller radii, signaling that the function has singularities closer to the origin.

By understanding how these numeric patterns reflect analytic structure, you gain insight into the behavior of complex functions far beyond mere series manipulation. This calculator aims to bridge numeric experimentation and theoretical intuition.

Root vs. Ratio Test in Practice

The root test uses the growth rate of individual coefficients, while the ratio test examines how consecutive coefficients relate. In well-behaved series both tests agree, but numerical estimates can differ when coefficients fluctuate. The calculator reports both so you can see how stable your data appears.

Boundary Behavior

The radius marks where absolute convergence fails, yet series may still converge conditionally on the boundary. Exploring values at $\left|z\right|=R$ often reveals subtle phenomena such as alternating or logarithmic divergence. Understanding boundary behavior is essential when performing analytic continuation.

Example Radii

Series	Coefficients	Radius
ez	1/n!	∞
1/(1−z)	1	1
ln(1+z)	((−1)^{n+1})/n	1

Singularities and Analytic Continuation

The nearest singularity of the underlying function dictates the radius. If a function has a pole at $z$₀, then $R=|z\_0|$. Recognizing this link guides analytic continuation: by re-centering a series around a new point beyond the original radius, one can often extend the function’s domain.

Tips for Accurate Estimates

Use as many coefficients as possible and simplify fractions before entering them. Outliers or zero coefficients can distort both tests, so review the sequence for consistency. When in doubt, plot the partial sums to visually inspect convergence.