Show that
nC1 + nC5 + nC9 + ... = 1/2 (2n-1 + 2n/2 sin (n pi/4))
2. (a) Show that the series sin "2n Sman 1 ) converges n = 1 (b) Find an estimate of the magnitude of the error if the sum of the series is calculated by summing up the first 20 terms of the series. [4+3=7 pts]
Recall that, for all c, = n=0 cos(x) = § 4 (-1)",21 (2n)! and sin(x) = (-1)"..2n+1 (2n + 1)! N=0 n=0 If i is defined to have the property that i = -1, show that ei cos(2) + isin(x) for any real number r.
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
For the set of functions {sin(x),sin(2x),sin(3x),...}=sin(nx)}, n=1,2,3,... on the interval [0,pi]. Show that the set of functions is orthogonal on [0,pi].
Entered Answer Preview Result (9/[(piÄ3)*(nA3)])*(16/(piA3)*(mA3)])* [pi*n'sin(pi*n)+2 cos(n*pi)-2 9 16 (πη sin(m) + 2 cos(n7)-2)(nm sin(mm) + 2 cos(mm)-2) incorrect 규 교 The answer above is NOT correct. 1 point) The double Fourier sine-sine series of the function is given by 00 oO m sin( ) sin(mm) where
Q11 (Variant of Wallis product). For every integer n 2 0, we define Im r sin dx (a) Show that In+,-n+21n (b) Show that 0< 12n+2 S I2n+i < 12n- (c) Use (a) and (b), show that lim Pni1. (d) Repeatedly using (a), show that I2nl2 (e) Compute the limit lim (Historical remark: Q9 and Wallis Product was one of earliest approaches to 22n(n)! 24n(n!)4 (2n+1)!(2n)! 2-n(n!)2- n→oo v 2n+1(2n)!' 22n (n!)2 and then lim approximate π from rationals. This...
2. Simplify: (n + 2)! (1) n! (2n-1)! (2) (2n + 1)! (2n + 2)! (3) (2n)!
For y = 3 sin 2(x - pi/4) Show your work! a) Find the amplitude, period, and horizontal shift of the function b) Graph the function.
1. Expand the following functions in terms of the orthogonal basis {1, sin 2nr. cos 2n on the interval (0, 1): n E Z, n > 0} 2. Expand the functions in problem i în terms of the basis {sin n z n є z,n > 0} on the interval (0, 1).
1. Expand the following functions in terms of the orthogonal basis {1, sin 2nr. cos 2n on the interval (0, 1): n E Z, n > 0} 2....
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...