Radius of Convergence & Interval of Convergence Calculator
Find the convergence domain for Power Series $\sum c_n(x-a)^n$ with Ratio Test derivation.
Supports:
n!, sqrt, ln, etc.Radius (R)
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Converges when $|x-a| < R$
Open Interval
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Check endpoints below
Center (a)
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Series Expansion Point
The Principle: Ratio Test Limit & Radius
1. Coefficient Ratio Limit ($L$)
$$ L = \lim \left| \frac{c_{n+1}}{c_n} \right| $$
2. The Radius Boundary ($R$)
$$ R = \frac{1}{L} $$
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L
R
Step-by-Step Ratio Test
To determine the radius, we calculate the limit of the absolute ratio of consecutive terms:
$$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
Step 1: Setup the Limit
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Step 2: Evaluate
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Step 3: Solve for $x$
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Endpoint Check Required
The Ratio Test determines the open interval. You must manually substitute the endpoints back into the original series to assume convergence ($\le$ or $\ge$).
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Left End
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Substituted Series:
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Right End
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Substituted Series:
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Professor’s Handbook
Mastering Convergence: The Ultimate Guide
A deep dive into the Ratio Test, endpoint analysis, and the logic behind infinite series convergence.
In my 20+ years of teaching Calculus II, finding the Interval of Convergence is consistently the topic where students lose the most points. Why? Because it requires a perfect storm of skills: limit evaluation, algebraic manipulation, and the dreaded “Endpoint Check.”
I designed this free Radius of Convergence Calculator to act as your personal tutor. It doesn’t just give you the answer; it visualizes the Ratio Test logic and prompts you to check the specific boundaries where standard tests fail.
1. The “Signal Tower” Analogy
Imagine a Power Series centered at $x=a$ as a radio tower. The signal is perfect at the center. As you move away, the signal degrades.
- Radius of Convergence ($R$): The maximum distance you can walk from the tower before you lose the signal completely. Inside this radius ($|x-a| < R$), the series is Absolutely Convergent.
- Interval of Convergence ($I$): The exact set of x-values, usually $(a-R, a+R)$, where the series is valid. This includes checking for Conditional Convergence at the edges.
🎓 The Golden Rule: The Ratio Test Formula
To find $R$, we almost always use the Ratio Test. We examine the limit of the absolute ratio of consecutive terms:
$$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
The series converges if $L < 1$. This leads directly to the radius formula $R = 1/(\text{Limit of Coefficients})$.
2. How to Find the Interval of Convergence
Whether you use our online solver or calculate by hand, the process for finding the interval always follows these standard steps.
Example Problem:
Find the interval of convergence for the series: $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n (x-5)^n}{n \cdot 4^n}$
Step 1: Apply the Ratio Test
We ignore the $(-1)^n$ (Absolute Convergence). We set up the limit:
$$ \lim_{n \to \infty} \left| \frac{(x-5)^{n+1}}{(n+1)4^{n+1}} \cdot \frac{n 4^n}{(x-5)^n} \right| = \frac{|x-5|}{4} $$
Step 2: Solve for the Radius ($R$)
Set the result $< 1$ for convergence: $\frac{|x-5|}{4} < 1 \implies |x-5| < 4$.
Result: Center $a=5$, Radius $R=4$. The preliminary interval is $(1, 9)$.
Step 3: The Endpoint Check (Critical!)
This is where the Endpoint Analysis feature of our calculator shines. We must plug $x=1$ and $x=9$ back into the original sum.
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At $x=9$: $\sum \frac{(-1)^n}{n}$. This is the Alternating Harmonic Series. It converges. Use
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At $x=1$: $\sum \frac{1}{n}$. This is the Harmonic Series (p-series with p=1). It diverges. Use
(.
Final Answer: $(1, 9]$
3. Absolute vs. Conditional Convergence
Understanding these terms is crucial for AP Calculus and University exams.
| Convergence Type | Where it happens | Series Test Used |
|---|---|---|
| Absolute Convergence | Strictly inside the interval ($|x-a| < R$). | Ratio Test / Root Test |
| Conditional Convergence | Usually at the Endpoints. | Alternating Series Test (AST) |
| Divergence | Outside the interval ($|x-a| > R$). | n-th Term Divergence Test |
4. Frequently Asked Questions (FAQ)
Q: Can I use the Root Test instead of the Ratio Test?
Yes. A Root Test Calculator solves limits using $\sqrt[n]{|a_n|}$. It is superior for series with $n$-th powers like $(a_n)^n$, but for factorials ($n!$), the Ratio Test (used by this tool) is the standard method.
Q: What is the radius of convergence for e^x, sin(x), and cos(x)?
For $e^x$, $\sin(x)$, and $\cos(x)$, the limit of the ratio is 0. Since $R = 1/0$, the radius is Infinity ($\infty$). These series converge for all real numbers.
Q: How do I handle n! (factorials) in the calculator?
Simply type n! in the input box. Our Power Series Calculator automatically handles factorial simplification (e.g., $(n+1)!/n! = n+1$) to determine the correct radius.
5. References & Authoritative Sources
For rigorous proofs and more practice problems, I recommend these standard resources:
1. Stewart, J. (2015). Calculus: Early Transcendentals.
Chapter 11.8: “Power Series”. The gold standard for understanding Radius of Convergence.
Chapter 11.8: “Power Series”. The gold standard for understanding Radius of Convergence.
2. Paul’s Online Math Notes (Lamar University)
A treasure trove of solved problems regarding Ratio Test failures and endpoint testing.
Visit Paul’s Notes →
A treasure trove of solved problems regarding Ratio Test failures and endpoint testing.
Visit Paul’s Notes →
3. Wolfram MathWorld
Advanced definitions for the Cauchy-Hadamard theorem ($R = 1 / \limsup \dots$).
Visit MathWorld →
Advanced definitions for the Cauchy-Hadamard theorem ($R = 1 / \limsup \dots$).
Visit MathWorld →
Check Your Series Convergence Now
Stop guessing. Get the radius, interval, and step-by-step derivation instantly with our free calculator.
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— Dr. Math (GoCalc Contributor), PhD in Applied Mathematics.
Making infinite series finite and understandable.
Making infinite series finite and understandable.