∑ Interval of Convergence Calculator with Steps – Free Power Series Solver 2026

What Is the Interval of Convergence? A Clear Explanation

When you work with a power series, the fundamental question is: for which values of x does the series actually add up to a finite number? The answer is the interval of convergence — the complete set of x-values where the series converges. Understanding this concept is essential in Calculus II, AP Calculus BC, and any course involving infinite series.

Every power series is centred at some value c and has the general form Σ aₙ(x−c)ⁿ. At x = c, every term is zero except the first, so the series always converges at its centre. Moving away from c, the series may converge (produce a finite sum) or diverge (blow up to infinity). The key tool for determining this is the Ratio Test, which gives you the radius of convergence R — the distance from the centre within which the series absolutely converges. The interval of convergence then extends from c−R to c+R, but the behaviour at the two endpoints requires separate testing.

How to Find the Interval of Convergence of a Power Series: The Method

The process of finding the interval of convergence always follows the same systematic steps. Once you understand the procedure, you can apply it to any power series — and this is exactly what the calculator above does, showing every step explicitly.

Step 1: Apply the Ratio Test

The Ratio Test is the standard tool for finding the radius of convergence. Compute the limit of the absolute value of the ratio of consecutive terms:

The Ratio Test tells us: if L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive and we need more information.

Step 2: Set Up the Inequality and Find the Radius

After computing the limit L (which will be an expression involving |x−c| divided by some constant), set L < 1 and solve for |x−c|:

Step 3: Check the Endpoints Individually

The Ratio Test gives L = 1 exactly at the endpoints x = c−R and x = c+R, so it cannot determine convergence there. Each endpoint must be tested separately by substituting into the original series and using a different convergence test. Common tests at endpoints include:

• Alternating Series Test — if the endpoint gives an alternating series that decreases to zero, it converges conditionally
• p-Series Test — if the endpoint gives Σ 1/nᵖ, it converges if p > 1 and diverges if p ≤ 1
• Divergence Test — if the terms don't approach zero, the series diverges
• Direct comparison with known convergent or divergent series

Step 4: Write the Interval Using the Correct Bracket Notation

Once you know whether each endpoint converges or diverges, write the interval:

Radius of Convergence Calculator with Steps: How R Is Computed

The radius of convergence R is the most important number associated with a power series. Here are the three possible outcomes and what they mean:

• R = ∞: The series converges absolutely for every real number x. This happens when the limit in the Ratio Test equals zero for any fixed x — meaning the factorials in the denominator (for series like eˣ, sin(x), cos(x)) grow so much faster than any xⁿ that the ratio goes to zero. These series converge everywhere on (−∞, +∞).
• R = a finite positive number: The series converges absolutely for x in (c−R, c+R). This is the most common case in textbooks and exams — geometric series, logarithm series, arctan series, and many others have a finite radius.
• R = 0: The series converges only at the single point x = c. This happens when the coefficient aₙ grows faster than any geometric sequence — for example, when aₙ = n! (factorial coefficients), the series n!·xⁿ diverges for any x ≠ 0.

Interval of Convergence for eˣ and Other Common Maclaurin Series

The most frequently tested series in Calculus II all have clean intervals of convergence. Understanding why they converge on their specific intervals — not just memorising the answers — makes you genuinely effective at solving unfamiliar problems.

Interval of Convergence for eˣ

The Maclaurin series for eˣ is one of the most beautiful results in mathematics: eˣ = Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + x⁴/4! + ... Applying the Ratio Test:

The key intuition is that factorials grow far faster than any exponential — n! eventually dwarfs xⁿ for any fixed x, no matter how large. This is why the exponential, sine, and cosine series (all involving factorials in the denominator) converge everywhere.

Interval of Convergence for Geometric Series Σ xⁿ

The geometric series Σ xⁿ = 1 + x + x² + x³ + ... is the simplest non-trivial example. Its sum formula is 1/(1−x), and the interval of convergence is exactly (−1, 1) — the series converges to 1/(1−x) precisely when |x| < 1. Applying the Ratio Test:

Interval of Convergence for ln(1+x)

The series for ln(1+x) = Σ (−1)ⁿ⁺¹xⁿ/n = x − x²/2 + x³/3 − x⁴/4 + ... has a more interesting endpoint story. The Ratio Test gives R = 1, and checking the endpoints:

This is a perfect example of why endpoint checking matters. The interval is neither fully open nor fully closed — it includes the right endpoint but not the left.

Interval of Convergence for arctan(x)

The series arctan(x) = Σ (−1)ⁿx²ⁿ⁺¹/(2n+1) = x − x³/3 + x⁵/5 − x⁷/7 + ... has the nicest endpoint behaviour of the common series. Both endpoints x = ±1 give alternating series that converge by the Alternating Series Test:

This result is used in the famous Leibniz formula for π: 1 − 1/3 + 1/5 − 1/7 + ... = π/4, which is arctan(1) evaluated using its convergent power series.

Interval of Convergence for Maclaurin Series: Complete Framework

A Maclaurin series is a Taylor series centred at c = 0. The convergence properties of Maclaurin series follow three distinct patterns depending on the type of function:

Pattern 1: Converge Everywhere — R = ∞

Functions whose Maclaurin series involve factorials in the denominator always converge everywhere. The factorial growth rate overwhelms any power of x. This group includes: eˣ, e⁻ˣ, sin(x), cos(x), sinh(x), cosh(x), and any composition of these (like sin(x²) or e^(x³)).

Pattern 2: Converge on (−1, 1) with Endpoint Variation — R = 1

Many fundamental functions have their Maclaurin series converge on (−1, 1) with varying endpoint behaviour. This group includes 1/(1−x), 1/(1+x), ln(1+x), arctan(x), arcsin(x), and the binomial series (1+x)^k. The open/closed nature of the interval at each endpoint must be determined individually using the Alternating Series Test or p-Series Test.

Pattern 3: Converge Only at the Centre — R = 0

Series with factorial or super-exponential coefficients in the numerator (like n!·xⁿ or (2n)!·xⁿ) have radius of convergence zero. They converge only at x = 0, which makes them essentially useless as computational tools for approximating function values away from the centre.

The Ratio Test Explained: Why It Works for Finding Convergence

The Ratio Test works by comparing consecutive terms of the series. If each successive term is a constant fraction of the previous one (like a geometric series), the series converges. The key insight is the limiting ratio L = lim|aₙ₊₁/aₙ|:

• If L < 1: each term is eventually smaller than r × (previous term) for some r < 1, making the series behave like a convergent geometric series
• If L > 1: each term is eventually growing, so the terms don't approach zero and the series must diverge
• If L = 1: the test is inconclusive — the series might converge or diverge, and we need a different approach

For power series, computing the Ratio Test always simplifies nicely because the xⁿ factors cancel, leaving an expression purely in terms of the coefficient pattern aₙ and the variable |x−c|. Setting this expression < 1 gives the radius of convergence directly.

Finding the Interval of Convergence: Worked Examples

Example 1: Σ xⁿ/n² — Ratio Test with p-Series Endpoints

Example 2: Σ n·xⁿ — Coefficients with n Factor

Example 3: Σ (x−2)ⁿ/3ⁿ — Non-Zero Centre

Interval of Convergence vs. Radius of Convergence: What's the Difference?

Students often conflate these two related concepts, so it's worth being precise. The radius of convergence R is a single non-negative number (or ∞) that tells you how far from the centre the series converges. It doesn't tell you about endpoint behaviour — it only describes the open interval (c−R, c+R) of absolute convergence.

The interval of convergence is the complete set of x-values where the series converges — it includes the radius information plus the endpoint determination. It may be open (−1, 1), half-open (−1, 1], or closed [−1, 1] depending on what happens at x = c±R. In other words: the radius gives you the size, and the interval gives you the full picture including boundaries.

Frequently Asked Questions: Interval of Convergence

Can the interval of convergence be a single point?

Yes. When R = 0, the interval of convergence is just {c}, the single centre point. This happens for power series with factorial coefficients in the numerator, like Σ n!·xⁿ. In this case, the series converges at x = 0 (every power of 0 is 0) but diverges for every x ≠ 0.

Is the interval of convergence always an interval?

Yes — this is a theorem in real analysis. The set of convergence for a power series Σ aₙ(x−c)ⁿ is always of one of three forms: just the centre {c}, an interval of positive length (open, half-open, or closed), or the entire real line (−∞, +∞). It can never be a disconnected set like the union of two separate intervals.

What does the interval of convergence mean geometrically?

Geometrically, the interval of convergence is the set of x-values where the partial sums of the power series actually approach the function the series represents. Outside this interval, the partial sums explode to ±∞ rather than converging to a finite value. For the series of 1/(1−x), for example, the partial sums 1 + x + x² + ... + xⁿ converge to 1/(1−x) only when |x| < 1 — outside this range, successive partial sums oscillate wildly or grow without bound.

Can I use the Root Test instead of the Ratio Test?

Yes. The Root Test (Cauchy-Hadamard formula) gives R = 1/lim(n→∞)|aₙ|^(1/n). It is particularly useful when the coefficient aₙ involves nth powers or nth roots, and it always gives the same radius as the Ratio Test when both tests are applicable. For series with factorial terms, the Ratio Test is typically simpler to compute. The choice of test doesn't affect the answer — only the computational difficulty.

What is the interval of convergence of a Taylor series?

A Taylor series centred at c has exactly the same interval of convergence structure as any other power series. The only difference from a Maclaurin series (centred at 0) is the shift: the interval is (c−R, c+R) instead of (−R, R). The steps for finding it are identical: apply the Ratio Test to find R, then check the endpoints x = c−R and x = c+R. For example, the Taylor series for ln(x) centred at c = 1 has interval of convergence (0, 2], and for 1/x centred at c = 1 has interval of convergence (0, 2).