Finite Sums and Infinite Series

Finite sums

Enter a term in the variable n and two integer bounds. Both bounds are included. Summing n from 1 through 100 gives 5050 because the 50 pairs from opposite ends each total 101. Use a decimal point, ln for natural logarithms and ^ for powers. Trigonometric functions use radians.

Infinite series

For the geometric family a·r^(n−1), |r|<1 gives a sum of a/(1−r). A zero initial term makes the entire series zero. The telescoping option is specifically 1/(n·(n+1)), whose sum is 1. The p series 1/n^p converges for p>1. That mode reports convergence but does not evaluate the infinite sum.

Numerical evidence is not proof

The general expression mode makes no convergence or divergence decision. The table gives sums of all terms up to selected indices and the absolute ratio of the last two terms. The ratio test requires a limit. A finite ratio is not a limit. Slowly growing sums such as the harmonic series can appear stable. The alternating harmonic series converges even though its absolute term ratio approaches 1. Do not treat a partial sum as an infinite sum. At most 100000 terms are evaluated, with no bound on the remaining tail.

Frequently asked questions

What is the sum of the integers from 1 through 100?

For example, summing the integer terms n from 1 through 100, including both limits, gives 5050. Pair the first and last terms in the arithmetic sum and multiply the common pair sum by the number of pairs, noting that this finite total does not establish convergence of an infinite series.

Calculate with these values