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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.
דוגמה עובדת · sum of 2^k for k = 0 to 10
צעד אחר צעד
- \sum_{k=0}^{10} 2^{k}
Write the sum out.
- = 2047
Closed form.
גלה את התשובה
עכשיו אתה בחר בעיה, או הקלד או צייר בעצמך. כל צעד, תמונה, התשובה חבויה עד שתשאל.
סמלים המשמשים כאן
הקש כל סמל להגדרה המלאה, תמונה, ומה כל אות בזה אומר.
איך: Sequences and series
- Write the sum out.
- Closed form.
שאלות שאנשים שואלים
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.
יותר בפנים. Precalculus
Complex numbersPolynomial functionsRational functionsThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates