Sum of Series Calculator

This calculator relies on foundational mathematical formulas depending on whether your sequence is arithmetic or geometric. For an arithmetic progression, the sum of n terms relies on the first term (a) and the common difference (d): S_n = (n / 2) * [2a + (n - 1)d]. For a finite geometric series, the total depends on the common ratio (r): S_n = a * (1 - r^n) / (1 - r). Finally, for an infinite geometric series to converge and yield a valid sum, the absolute value of the common ratio must be strictly less than 1 (|r| < 1), governed by the elegant limit formula S = a / (1 - r).

Imagine you need to find the total sum of an arithmetic sequence where the first term (a) is 5, the common difference (d) is 3, and the total number of terms (n) is 10. First, substitute these values into the arithmetic sum formula: S_10 = (10 / 2) * [2(5) + (10 - 1)3]. Simplifying inside the brackets gives us [10 + (9 * 3)], which equals 10 + 27 = 37. Next, multiply by the outside coefficient (10 / 2 = 5). Finally, 5 * 37 yields a grand total sum of 185 for your series.

Always verify whether your geometric series has a common ratio whose absolute value is less than one before attempting to calculate an infinite sum, as ratios greater than or equal to one cause divergence to infinity. Double-check your term count (n); forgetting to subtract one from n in the arithmetic formula is a frequent source of calculation errors. When dealing with mixed sequences, break them down into distinct arithmetic and geometric components before running individual summations.