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Sequences and series

Arithmetic and geometric sequences, partial sums, and the geometric series formula.

An arithmetic sequence adds the same amount each step; a geometric one multiplies by the same ratio. Their sums have closed forms, n(a₁ + aₙ)/2 and a(rⁿ − 1)/(r − 1), and when |r| < 1 the geometric series keeps converging as n grows, which is the first infinite sum most people meet.

Àwọn ààtò àwọn ìṣàmúlò-ètò · sum of 2^k for k = 0 to 10

Calculus · Series-And-Sums

Sum 2^k for k = 0 to 10

\sum_{k=0}^{10} 2^{k}

Àwọn ìṣàmúlò-ètò

\sum_{k=0}^{10} 2^{k}
Write the sum out.
= 2047
Closed form.

Fi àwọn àgbèwọlé hàn

2047

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Bii o ṣe le: Sequences and series

Write the sum out.
Closed form.

Àwọn Àtòjọ-ẹ̀yàn

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Diẹ̀ nínú Precalculus

Complex numbers Polynomial functions Rational functions The binomial theorem Conic sections Vectors Exponential and logarithmic functions Polynomial division and the remainder theorem Parametric equations and polar coordinates