How to Calculate the Interval of Convergence

How to Calculate Interval of Convergence: Step-by-Step Guide (2026)

Understanding how to find the interval of convergence by hand is essential for mastering power series. While our Interval of Convergence Calculator automates the process, manual calculation builds deeper intuition. This guide walks you through every step with clear examples.

What You'll Need

• Pen and paper
• Basic calculus knowledge (limits, series)
• The power series you want to analyze
• Convergence test formulas (ratio test or root test)

Step-by-Step Process

1. Identify the Series and Center
   Write the power series in the form Σ aₙ(x - c)ⁿ. Determine the center c and the coefficient pattern aₙ.
2. Choose a Convergence Test
   Usually the ratio test (L = lim |aₙ₊₁/aₙ|) or root test (L = lim (|aₙ|)^(1/n)). The calculator uses these; you’ll do the same manually.
3. Compute the Limit L
   Apply the chosen test to find the limit as n→∞. This limit depends only on the coefficients, not on x. For example, for aₙ = 1, L=1.
4. Find the Radius R
   Radius R = 1/L. If L=0, R=∞; if L=∞, R=0. Otherwise, R = 1/L.
5. Write the Open Interval
   The series converges absolutely for |x-c| < R, i.e., c - R < x < c + R. This is the open interval.
6. Test the Endpoints
   Plug x = c - R and x = c + R into the original series. Determine convergence using p-series, alternating series test, etc. Endpoints may be included (converges) or excluded (diverges).
7. State the Final Interval
   Combine the open interval with endpoint results. Example: (c-R, c+R], [c-R, c+R), (c-R, c+R), or [c-R, c+R].

Worked Examples

Example 1: Σ (xⁿ)/(n!) centered at 0

Here, aₙ = 1/n! and c=0. Use ratio test: lim |aₙ₊₁/aₙ| = lim | (1/(n+1)!) / (1/n!) | = lim 1/(n+1) = 0. So L=0, R=∞. The open interval is (-∞, ∞). Since endpoints are not finite, the interval is (-∞, ∞).

Example 2: Σ (x-2)ⁿ / n² centered at 2

aₙ = 1/n², c=2. Ratio test: lim |aₙ₊₁/aₙ| = lim n²/(n+1)² = 1. So L=1, R=1. Open interval: (1,3). Test endpoints: x=1 gives Σ (-1)ⁿ/n² (converges by alternating series test). x=3 gives Σ 1/n² (converges as p-series with p=2>1). Hence interval: [1,3].

Common Pitfalls

• Forgetting absolute values: The ratio and root tests always use absolute values. Omitting them leads to wrong radius.
• Misapplying tests at endpoints: The tests only give convergence inside the open interval; endpoints require separate analysis.
• Ignoring the center: The interval is symmetric around c, not necessarily around 0.
• Algebra errors: When simplifying limits, careful with factorials and powers.

For a deeper look at the formula and test details, check our formula guide. If you're working with Taylor series, see our Taylor series instance page. For common questions, visit the FAQ page.