Mathematics · 34 visual lessons

Sequences and infinite series
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Explore integration, differential equations, series, and geometry through physical models.

Follow the real situation, derive the mathematics, and test it with interactive graphs and worked practice.

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Unit 4 · 9 lessons

Sequences and infinite series

Repeated approximations can settle toward a target. The index labels discrete observations rather than continuous time.

A repeating fractional contribution can add to a finite total. Follow the partial sums instead of only the individual contributions.

A decreasing positive sequence of contributions can be bracketed by areas. The same construction also bounds the remainder.

A complicated positive term may be bounded by a simpler power-law term. Use that inequality to decide convergence.

Terms involving factorials can initially grow, yet eventually shrink rapidly. Inspect how one term compares with the next.

An nth root can separate exponential growth from slower polynomial effects. Its limiting value decides most cases.

Positive and negative contributions can settle even when the sum of their magnitudes grows without bound.

A finite polynomial can be evaluated anywhere, but an infinite series may represent a function only inside a specific radius.

An exponential response can be approximated near zero by a line, a quadratic, or a higher-degree polynomial. Compare error as you change degree.