Show that — ф+1 n=0 ||

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 1. Find the Fourier series of the function: f(x) = |x| in the period of -n<x<

A: Fourier series formulae: f(x)=a02+∑n=1∞ancos(nx)+bnsin(nx)a0=1π∫-ππf(x)dxan=1π∫-ππf(x)…

Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: 9.5 The celebrated golden ratio o is the positive real number number satisfying 1/o = $ – 1. Show…

A:

Q: Choose {Xi}ni=1 from [0, 1]. Let pn be the probability that Xi + Xi+1 ≤ 1 for all i ∈ {1, 2, . . . ,…

A: Given that : Xi∈0,1 pn=pXi+Xi+1≤1   ∀  i=1,2,3,4...n-1 We have to calculate limn→∞pn1n PXi+Xi+1≤x…

Q: The results of a mathematics exam at JUST university for two cla Class 1: (n=32, mean=15.7, SD=8)…

A: Given,n1=32 , μ1=15.7 , σ1=8n2=36 , μ2=4.5 , σ2=12α=1-0.95=0.05α2=0.025Z0.025=1.96

Q: ,2 COS“ N cos“ n In n n2 n3 n=1

A:

Q: Suppose that the Fourier series expansion for function { 2 cos x, if - Ħ< x < 0 f (x) = if 0 < x < T…

A: Given fx=2cosxif -π&lt;x&lt;0sinxif  0&lt;x&lt;π

Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

A:

Q: 3. Let n E Z be odd. Show that there exist two consecutive integers such that their sum is equal to…

A: Given n be a odd integer. Which is of the form n= 2k+1, k is an integer.

Q: Expand cos'x as a series of ascending powers of x.

A: We have to expand cos2x as a series of ascending power of x. Recall that:           cos2x =2cos2x -1…

Q: ΑΣ 4. Suppose you know that f(x) = |æ| = 5 - specialization method to show that +++ makes the sum El…

A: Fourier series expansion of a function defined in -π,π is given by, fx=a02+∑n=1∞ancosnx+bnsinnx…

Q: @) 2) Fs of (x) = -PFC (Psy) -} [ If f (x,y) → as 6004X proof. (H-w.) then в

A: Introduction: The Fourier sine and cosine transforms are variations of the Fourier rework that do…

Q: R² = gen gen

A: see below the reason

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

A:

Q: Assume that after w months,butterfiles populace in 1000 is calculated as F(w)=12w+cos(16w)+n×100…

A: Minimum and maximum of a function depends upon all the variables present in the function.

Q: 2. (a) Use Taylor's theorem to find the first degree expansions of (i) sin x, (ii) e and (iii)…

A: for the function  f(x)  taylor polynomail of first degree about  x=0 is p(x)=f(0)+x f'(0)

Q: Find the Fourier series representation of the odd extension of the function. Then, find the value of…

A: Fourier series for the odd extension of a function defined in (0,L) is given by fx=∑n=1∞ bn…

Q: 3. The Chebyshev polynomials of the first kind are defined as Tn(x) = = cos (n arccos(x)) (b) Show…

A: in the given question we have to prove that the chebyshev polynomial Tn(x) = cos(ncos-1(x)).…

Q: 2. Show that X H→ In x is increasing on (0,0) and that In 1 = 0:

A:

Q: 3, Re call Eudlid's proof of In finitude * primes! Shouw that T Cx): = 1{ ps X: p-pinae }12 logz…

A:

Q: in ascending powers of x, up to and including the 4th term. Hence, Expand (6 – approximate (5.6)6.…

A: To expand the given expression.

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: 8 find Taylor's Beries for In (x+1) about x=. f(x) = du (x +).

A: Taylor series of a function fx about a point x=a  is given by  fx=∑k=0∞fkax-ak     ,   fka denotes…

Q: Prove Bernoulli's Identity: For every real numberx > -1 and every positive integer n, (1+x)" > 1+nx.

A:

Q: 12. Find all possible values of x such that In(x) + ln(x + 1) = In(2). If no such values of x exist,…

A:

Q: Suppose that phi (x) belongs to Aut (Zn) and a is relatively prime to n.  If phi (a)=b, determine a…

A: According to the given information it is given that:

Q: 21. Find the Fourier r fe -1<x <o - -IN X

A:

Q: The interval [0,5] is partitioned into nn equal subintervals, and a number xi is arbitrarily chosen…

A: The given interval is [0,5]. Find the xi.

Q: 99. Show that 00 cos x dx is absolutely convergent.

A:

Q: An ancient technique for extracting the square root of an integer N > 1 going back to the…

A:

Q: Use Heaviside expansion theorem (4s^2+7s+1)/s(s+1)^2

A:

Q: Determine the limit of the sequences (2 * n + 1)2 – (n – 1)2 an = (n + 1)2 – (n – 1)2 cos(42 * n3)…

A:

Q: Solve for x. All final answers must be in Zn where n is the modulus. 6x mod 11 ≡1

A: Solving this is equivalent to finding multiplicative inverse of 6 in  Z11 . We have to find x such…

Q: A/ Show that as a half range series of cosine terms for 0 < x < 1, π² /cos 2x cos 4x cos 6x +- 6 +…

A:

Q: use Hew Find the natureof the fallowingsekies 3-s ncn+i) nel hth=termtest- { Ln to to en 4- n² +3 en…

A: we can answer only three subparts, or the first question and the rest can be reposted. No more than…

Q: The interval [0, 4] is partitioned into n equal subintervals, and a number x, is arbitrarily chosen…

A:

Q: Let the characteristic of F be n =/0 . Then n is prime.

A:

Q: 3) Find the Term that includes x? of the expansion (x +

A:

Q: 3. Expand cos'x as a series of ascending powers of x. OS

A: To expand cos^2x as a series of ascending powers of x.

Q: Sigma(n=1 to n= infinity) 5* (2n-

A: We have to solve given problem:

Q: 5. Prove that . Hint: Compute e-' from the continued frac e - 1 1 + 2 + 3 + ·…. tion for et given in…

A: When a series is in the form ∑k=1∞-1k-1α1α2α3...αk, we can express it as a continued fraction as…

Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

Q: (Xn) be a sequence of real numbers defined recursively by 1 = 1 and 1 7. Let X = In+1 =5(Tn +3) for…

A:

Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

A:

Q: Using binomial expansion, prove that z3 = [r (cos 0 + isin 0)]3 is equal to r3 (cos 30 + isin 30).

A:

Q: 5. Show that cos .3 G) do not converge.

A: We need to prove that: cosnπ3 does not converge This can be proved as follows:

Q: Suppose the interval [1,9] is divided into n subintervals of equal width, and Ck represents the left…

A:

Q: 6) Use the first three terms of the Taylor expansion for f(x) = cos(x) to estimate cos(1/8). %3D

A: We have to find the value of cos(1/8)using only first three terms of Taylor's series.

Q: Let 0 < 0 < 1 be a rational number. Show that the sequence {cos(nT0)}-1 is divergent.

A: