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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.
Contoh yang dikerjakan · sum of 2^k for k = 0 to 10
Langkah demi langkah
- \sum_{k=0}^{10} 2^{k}
Write the sum out.
- = 2047
Closed form.
Tunjukkan jawapan
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Tekan simbol mana-mana untuk definisi penuh, gambar, dan apa makna setiap huruf dalamnya.
Bagaimana untuk: Sequences and series
- Write the sum out.
- Closed form.
Soalan yang orang tanya
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.
Lebih dalam Precalculus
Complex numbersPolynomial functionsRational functionsThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates