Convergent Sequence Calculator

Six convergent series; two irrational

Here are the six sequences that can be calculated to approximate the answer upon which they converge:

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 2

1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots = \frac{2}{3}

1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots = \frac{3}{2}

1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \dots = \frac{3}{4}

\frac{2 \times 2}{1 \times 3} \times \frac{6 \times 6}{5 \times 7} \times \frac{10 \times 10}{9 \times 11} \times \frac{14 \times 14}{13 \times 15} \times \dots = \sqrt[2]{2}

\frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \dots = \frac{π}{2}

The fifth series is a little less obvious as far as constructing the terms. Each term is constructed as follows with k starting at 0:

\frac{{(4 k + 2)}^{2}}{(4 k + 1) (4 k + 3)}

The final series is the Wallis formula for π/2. A beautiful graphic proof of this series' equality with π/2 was presented by the 3Blue1Brown youtube channel in 2018. This calculation actually computes a pair of terms (with the same numerator) in the series versus a single term in the series for all of the others. (e.g. if ten is enter for "Number of Terms", 20 terms are calculated.)

The calculator works as follows: select a convergent series from the drop down menu, enter the number of terms for your approximation, and hit the "Calculate" button. The output shows the number calculated and the difference from the limit.

Enter a number and hit the button: Select the sequence:
Number of terms

Approximation =
Difference =