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Series Formulas 1. Arithmetic and Geometric Series Definitions: First term: a 1 Nth term: a n Number of terms in the series: n Sum of the first n terms: S n Difference between successive terms: d Common ratio: q Sum to infinity: S Arithmetic Series Formulas: a a n dn = + −1 (1) 1 1 2 i i i a a a − + + = 1 2 n n a a S n + = ⋅ 2 11 ( ) n 2 ...
This section explains three Fourier series: sines, cosines, and exponentials eikx. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative. We look at a spike, a step function, and a ramp—and smoother functions too.
Mathematics: applications and interpretation formula booklet 2 . Prior learning – SL and HL Area of a parallelogram is the heightA bh=, where b is the base, h Area of a triangle , whe. 1 2 A bh= b is the base, re h is the height . Area of a trapezoid . 1 2. A a bh = +, where .
Finite orthogonal series b = c1v1 +c2v2 +···+cnvn. (13) Multiply both sides by vT 1. Use orthogonality, so vT1v2 = 0. Only the c term is left: Coefficient c1 vT 1 b = c1v T 1 v1 + 0+···+0. Therefore c1 = vT1b/vT 1 v1. (14) The denominator vT 1 v1 is the length squared, like π in equation (11). The numerator vT
Given that functions are determined by their Fourier coefficients, we wish to find formulas for them using their Fourier coefficients. Denote the partial sums, Ces ́aro means, and Abel means, respectively, of the Fourier series by. N → ∞, and maxx |A(x, r) − F (x)| tends to 0 as r → 1 (r < 1).
Fourier Analysis is the process of nding the spectrum, Xk, given the signal x(t). I'll tell you how to do that today. . . . but we don't know a, b, c, d, etc. Let's use orthogonality to gure out the value of b: nd the value of b from the formula above. We still have one problem.
1. (a) Show that the following sequences are arithmetic. (b) Find the common difference. (c) Define the rule that gives the nth term of the sequence. i. {2, 6, 10, 14, . . . } ii. {20, 17, 14, 11, . . . } iii. { 1, –4, –9, . . . } iv. {0.5, 1.0, 1.5, 2.0, . . . } v. {y + 1, y + 3, y + 5, . . . } vi. {x + 2, x, x – 2, . . . } 2.
