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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.

Mfano wenye matokeo: 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}

Hatua kwa hatua

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

    Write the sum out.

  2. = 2047

    Closed form.

Lafunua jibu
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.

Jaribu kufanya mambo yako mwenyewe

Mengi zaidi katika Precalculus

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