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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.
Umzekelo osebenzelayo · sum of 2^k for k = 0 to 10
Inyathelo ngenyathelo
- \sum_{k=0}^{10} 2^{k}
Write the sum out.
- = 2047
Closed form.
Bonisa impendulo
ngoku Khetha ingxaki, okanye ubhale okanye uzobe yakho. Inyathelo ngalinye, umfanekiso, impendulo ifihlakele de ubuze.
I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.
Bhalisa Igama elithileIimpawu ezisetyenziswa apha
Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.
Indlela yokusebenza: Sequences and series
- Write the sum out.
- Closed form.
Imibuzo abantu bebuza
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.
IiNkqubo Precalculus
Complex numbersPolynomial functionsRational functionsThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates