0.009766
9.990234
10
1
99.9023
%
0.009766
0.009766
9.990234
10
1
99.9023
%
0.009766
The Geometric Series Calculator computes partial sums, infinite sums, and convergence analysis for geometric series — sequences where each term is a fixed multiple of the previous one. Enter the first term $$a_1$$, common ratio $$r$$, and number of terms $$n$$ to get comprehensive results.
A geometric series is the sum of a geometric sequence: $$a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \cdots$$ The defining feature is that each term is obtained by multiplying the previous term by the constant ratio $$r$$. When $$|r| < 1$$, the terms shrink toward zero and the series converges to a finite sum. When $$|r| \geq 1$$, the terms either stay constant, oscillate, or grow, and the series diverges.
The geometric series is arguably the most important series in mathematics. It is one of the few infinite series with a simple, exact closed-form sum: $$S_\infty = a_1 / (1 - r)$$ for $$|r| < 1$$. This formula underpins vast areas of mathematics and applied science. In calculus, geometric series are used to derive Taylor series and test convergence. In probability, the geometric distribution models the number of trials until the first success. In economics, discounted cash flow analysis uses geometric series to value streams of future payments.
The partial sum formula $$S_n = a_1 (1 - r^n)/(1 - r)$$ has a particularly elegant derivation. Multiplying the series by $$r$$ shifts every term one position, and subtracting the shifted series from the original cancels all but the first and last terms. This telescoping trick is one of the most beautiful arguments in elementary algebra.
Geometric series also illuminate the concept of convergence rate. The ratio $$S_n / S_\infty$$ tells you what fraction of the infinite sum you have captured after $$n$$ terms. For $$r = 0.5$$, ten terms capture 99.9% of the infinite sum. For $$r = 0.9$$, you need 44 terms to reach 99%. This calculator displays the ratio $$S_n / S_\infty$$ as a percentage, giving you an intuitive measure of how quickly the series converges.
Enter your parameters below to explore geometric series behavior. Experiment with different ratios to see how convergence speed varies dramatically with $$r$$.
Step 1: General term. The $$n$$-th term of a geometric sequence is:
$$a_n = a_1 \cdot r^{n-1}$$
Step 2: Partial sum. The sum of the first $$n$$ terms (when $$r \neq 1$$) is derived via the telescoping method:
$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$
When $$r = 1$$: $$S_n = n \cdot a_1$$.
Step 3: Infinite sum. If $$|r| < 1$$, then $$r^n \to 0$$ as $$n \to \infty$$, and the partial sum converges to:
$$S_\infty = \frac{a_1}{1 - r}$$
If $$|r| \geq 1$$, the infinite series has no finite sum (diverges).
Step 4: Convergence ratio. For convergent series, the calculator computes:
$$\text{Ratio} = \frac{S_n}{S_\infty} = 1 - r^n$$
This percentage shows how close the partial sum is to the infinite limit. It depends only on $$r$$ and $$n$$, not on $$a_1$$.
Partial Sum Sₙ is the sum of the first $$n$$ terms. For convergent series, this approaches $$S_\infty$$ from below (when $$0 < r < 1$$) or oscillates around it (when $$-1 < r < 0$$).
Last Term aₙ shows the magnitude of the $$n$$-th term. For convergent series, this value is very small, confirming that the tail of the series contributes little to the total.
Infinite Sum S∞ is the exact limit of the series when $$|r| < 1$$. It is displayed as 0 for divergent series, indicating no finite limit exists.
Convergent? gives a clear verdict with explanation. The boundary $$|r| = 1$$ is a critical threshold: below it the series converges; at or above it, the series diverges.
Sₙ/S∞ expressed as a percentage tells you how much of the total infinite sum your partial sum captures. Values near 100% indicate fast convergence.
Inputs
Results
S₁₀ = 5(1 − 0.5¹⁰)/(1 − 0.5) = 5 × 0.999023/0.5 = 9.990234. S∞ = 5/(1 − 0.5) = 10. After 10 terms, 99.9% of the infinite sum is captured. The last term a₁₀ = 5 × 0.5⁹ = 0.00977 is tiny.
Inputs
Results
S∞ = 100/(1 − 0.9) = 1000. After 20 terms, S₂₀ = 878.42 — only 87.8% of the limit. The ratio r = 0.9 is close to 1, so convergence is slow. You need about 44 terms to reach 99% of S∞.
For a series to converge, its terms must approach zero. In a geometric sequence, $$a_n = a_1 r^{n-1}$$. If $$|r| < 1$$, then $$|r^n| \to 0$$, so terms vanish. If $$|r| \geq 1$$, terms do not approach zero, and the necessary condition for convergence fails. The series diverges.
When $$r < 0$$, the terms alternate in sign: positive, negative, positive, negative, … This is called an alternating geometric series. If $$|r| < 1$$ it still converges, but the partial sums oscillate above and below $$S_\infty$$ rather than approaching it monotonically from one side.
Yes. A negative $$a_1$$ simply flips the sign of every term and the sum. The convergence behavior depends only on $$r$$, not on $$a_1$$.
This ratio tells you the fraction of the infinite total captured by the first $$n$$ terms. If you need 99% accuracy, set $$1 - r^n \geq 0.99$$ and solve for $$n$$: $$n \geq \log(0.01) / \log(|r|)$$. For $$r = 0.5$$, you need $$n \geq 7$$. For $$r = 0.99$$, you need $$n \geq 459$$.
Geometric series model compound interest, depreciation, drug dosage accumulation in pharmacology, signal attenuation in telecommunications, population growth (discrete model), and discounted cash flow in finance. Any process involving repeated multiplication by a constant factor produces a geometric series.
Zeno argued that to traverse a distance, you must first cover half, then half of the remainder, and so on — an infinite number of steps. The geometric series $$1/2 + 1/4 + 1/8 + \cdots = 1$$ shows that infinitely many steps can sum to a finite total. The distance is fully covered, resolving the paradox.
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