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The Sum of Series Calculator computes partial sums and convergence information for the four most important families of infinite series: arithmetic, geometric, harmonic, and p-series. Understanding series summation is fundamental to calculus, analysis, and applied mathematics.
An arithmetic series has a constant difference $$d$$ between consecutive terms. The $$n$$th term and partial sum are given by closed-form expressions:
$$a_n = a_1 + (n-1)d, \qquad S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$
Arithmetic series always diverge (they grow without bound) unless $$d = 0$$ and $$a_1 = 0$$.
A geometric series has a constant ratio $$r$$ between consecutive terms. Its partial sum and infinite sum (when convergent) are:
$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \qquad S_\infty = \frac{a_1}{1 - r} \quad \text{for } |r| < 1$$
The geometric series is perhaps the single most important series in mathematics—it appears in probability, finance (compound interest, annuities), signal processing (z-transforms), and fractal geometry.
The harmonic series $$\sum_{k=1}^{\infty} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots$$ is the famous divergent series whose terms approach zero yet whose partial sums grow without bound. The partial sum is approximated by:
$$H_n \approx \ln(n) + \gamma + \frac{1}{2n} - \frac{1}{12n^2}$$
where $$\gamma \approx 0.5772$$ is the Euler–Mascheroni constant.
The p-series $$\sum_{k=1}^{\infty} \frac{1}{k^p}$$ generalizes the harmonic series. It converges if and only if $$p > 1$$. The most famous case is the Basel problem: $$\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$$, solved by Euler in 1734.
Series summation appears throughout science and engineering—from computing net present values in finance, to analyzing signal attenuation in telecommunications, to calculating energy levels in quantum mechanics. The convergence or divergence of a series determines whether a physical quantity remains bounded or grows without limit.
Select the series type, enter the first term, the common difference or ratio, and the number of terms. For p-series, also specify the exponent $$p$$. The calculator returns the partial sum $$S_n$$, the last term $$a_n$$, and the infinite sum if the series converges.
The partial sum $$S_n$$ is the total after adding $$n$$ terms. The infinite sum (when shown as nonzero) indicates that adding all infinitely many terms gives a finite result. A value of 0 for the infinite sum means the series diverges—the partial sums grow without bound.
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The sum 3 + 1.5 + 0.75 + ... converges to S∞ = 3/(1−0.5) = 6. After 10 terms the partial sum is already 5.994, within 0.1% of the limit.
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S₂₀ = 20/2 × (5 + 62) = 10 × 67 = 670. The last term is a₂₀ = 5 + 19×3 = 62. The series diverges.
For geometric series, convergence requires $$|r| < 1$$. For p-series, convergence requires $$p > 1$$. Arithmetic series always diverge (unless trivially zero). The harmonic series ($$p = 1$$) diverges despite its terms approaching zero, illustrating that the terms going to zero is necessary but not sufficient for convergence.
A sequence is an ordered list of numbers $$a_1, a_2, a_3, \ldots$$. A series is the sum of a sequence: $$S = a_1 + a_2 + a_3 + \cdots$$. The partial sum $$S_n$$ is the sum of the first $$n$$ terms of the sequence.
The terms $$1/k$$ decrease too slowly. Grouping terms shows that each doubling of $$k$$ adds at least $$1/2$$ to the sum: $$1 + (1/2) + (1/3 + 1/4) + (1/5 + \cdots + 1/8) + \cdots \ge 1 + 1/2 + 1/2 + 1/2 + \cdots$$, which diverges.
The Euler–Mascheroni constant $$\gamma \approx 0.5772$$ is defined as $$\gamma = \lim_{n \to \infty}(H_n - \ln n)$$, where $$H_n$$ is the $$n$$th harmonic number. It measures how much the harmonic series exceeds the natural logarithm and appears throughout number theory and analysis.
The present value of a perpetuity paying $$C$$ per period at discount rate $$r$$ is $$PV = C/r$$, which is the infinite geometric sum with ratio $$1/(1+r)$$. Similarly, annuity formulas for mortgages and bonds are based on finite geometric sums.
A displayed infinite sum of 0 indicates that the series diverges—it has no finite limit. The partial sum will continue to grow as more terms are added. Only convergent series have meaningful infinite sum values.
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