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59,048
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The Series Calculator evaluates both arithmetic and geometric series, providing partial sums, the last term, and convergence analysis. Select the series type, enter the first term and common difference or ratio, and get instant results for any number of terms.
A series is the sum of the terms of a sequence. While sequences list individual values, series accumulate them. The distinction matters enormously in mathematics, physics, and engineering, where series appear in everything from calculating compound interest to approximating transcendental functions via Taylor expansions.
An arithmetic series sums terms that increase by a constant difference $$d$$. The $$n$$-th partial sum is $$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$, a formula attributed to the young Carl Friedrich Gauss, who reportedly derived it at age seven by pairing terms from opposite ends of the sum $$1 + 2 + \cdots + 100$$. Arithmetic series always diverge — the partial sums grow without bound (unless $$d = 0$$, giving a constant series).
A geometric series sums terms that multiply by a constant ratio $$r$$. The partial sum formula is $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ when $$r \neq 1$$. The critical distinction with geometric series is convergence: if $$|r| < 1$$, the series converges to the finite value $$S_\infty = \frac{a_1}{1 - r}$$. If $$|r| \geq 1$$, the series diverges. This convergence property makes geometric series fundamental in calculus, probability theory, and financial mathematics.
In finance, the present value of an annuity is a geometric series. In physics, repeated reflections between parallel mirrors or successive bounces of a ball form geometric series. In signal processing, the impulse response of many filters is a geometric series. In computer science, the analysis of divide-and-conquer algorithms often involves geometric series arising from the Master Theorem.
This calculator handles both types seamlessly. Choose arithmetic or geometric, provide the parameters, and receive the partial sum, final term, infinite sum (for convergent geometric series), and a clear convergence verdict.
Arithmetic Series
Given first term $$a_1$$, common difference $$d$$, and number of terms $$n$$:
The $$n$$-th term: $$a_n = a_1 + (n-1)d$$
The partial sum: $$S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)$$
This is equivalent to $$n$$ times the average of the first and last terms.
Geometric Series
Given first term $$a_1$$, common ratio $$r$$, and number of terms $$n$$:
The $$n$$-th term: $$a_n = a_1 \cdot r^{n-1}$$
The partial sum (when $$r \neq 1$$): $$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$
When $$r = 1$$: $$S_n = n \cdot a_1$$
Infinite Sum (Geometric Only)
If $$|r| < 1$$, as $$n \to \infty$$, $$r^n \to 0$$, so: $$S_\infty = \frac{a_1}{1 - r}$$
If $$|r| \geq 1$$, the infinite series diverges — no finite sum exists.
Partial Sum Sₙ is the total when you add the first $$n$$ terms. For an arithmetic series this grows quadratically with $$n$$; for a geometric series with $$|r| > 1$$ it grows exponentially.
Last Term aₙ is the value of the $$n$$-th term. Comparing it to $$a_1$$ shows how the sequence grows or decays.
Infinite Sum S∞ is displayed only for convergent geometric series ($$|r| < 1$$). It represents the value the partial sums approach as $$n \to \infty$$. For arithmetic series or divergent geometric series, this field shows 0 indicating no finite limit.
Convergent? tells you whether the infinite series converges. Arithmetic series never converge (except trivially when $$d = 0$$). Geometric series converge only when $$|r| < 1$$.
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The series is 1 + 3 + 5 + 7 + … + 19. Using the formula: S₁₀ = 10/2 × (2×1 + 9×2) = 5 × 20 = 100. This confirms the well-known result that the sum of the first n odd numbers equals n².
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The series is 8 + 4 + 2 + 1 + 0.5 + … S₁₀ = 8(1 − 0.5¹⁰)/(1 − 0.5) = 8 × 0.999023/0.5 ≈ 15.984. The infinite sum S∞ = 8/(1−0.5) = 16. After 10 terms, the partial sum is already within 0.1% of the limit.
A sequence is an ordered list of numbers (e.g., 2, 5, 8, 11, …). A series is the sum of the terms of a sequence (e.g., 2 + 5 + 8 + 11 + … ). The $$n$$-th partial sum of a series adds only the first $$n$$ terms.
An arithmetic series converges only in the trivial case where the common difference $$d = 0$$ (all terms are identical, but the sum still grows as $$n \cdot a_1$$, which diverges unless $$a_1 = 0$$). In general, arithmetic series always diverge because the terms do not approach zero.
When $$r = 1$$, every term equals $$a_1$$, so the partial sum is simply $$S_n = n \cdot a_1$$. The standard formula $$a_1(1 - r^n)/(1 - r)$$ has a zero denominator, so this special case is handled separately.
A series converges if its partial sums approach a finite limit as the number of terms increases without bound. For a geometric series, convergence occurs when $$|r| < 1$$ because each successive term is smaller than the previous one, and the total remains bounded.
Annuity calculations, loan amortization, and compound interest involve geometric series. For example, the present value of $$n$$ equal payments at interest rate $$i$$ is $$PV = PMT \cdot \frac{1 - (1+i)^{-n}}{i}$$, which is a rearrangement of the geometric series formula.
The meaning of the second parameter depends on the series type you select. For arithmetic series, it is the common difference $$d$$ (added to each term). For geometric series, it is the common ratio $$r$$ (multiplied by each term). The label adjusts contextually.
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