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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.

Apdorotas pavyzdys: 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}

Žingsnis po žingsnio

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

    Write the sum out.

  2. = 2047

    Closed form.

Atskleisti atsakymą
2047

Symbols used here

\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.

How to: Sequences and series

  1. Write the sum out.
  2. Closed form.

Questions people ask

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.

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Daugiau informacijos Precalculus

Complex numbersPolynomial functionsRational functionsThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates