Prove n (1+x)n-1= Σ c(n, k) k xk-1. k=1

Q: [(r* - 1) (4 – x* )ax

A:

Q: Bwalya was born in 1980. In two leap years to come, Shabalala was born. In which year was Shabalala…

A: Given, Bwalya was born in 1980. We know that 1980 is a leap year. Also, we have a leap year come…

Q: Show that ø(n²) = nø(n) for every n E Z*.

A: I have used only one formula of euler phi function

Q: bho edop Let S(n) denote the statement " 2i = n2 + n+1." %3D i=1 (a) Prove that the truth of S(k)…

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Q: 1' 1 3 (e) Show that 5 k=1 k k+2

A: I am attaching image so that you understand each and every step.

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: rove that (:)а K М— К (К):(М — К)п-k k п — k (М), ( M

A: It is an important part of statistics. It is widely used.

Q: 1 Q6) Let H(x):=k for x=, (k eN) and H(x):=0 elsewhere on [0,1] k Determine whether H(x) eR[0,1 or…

A: Unbounded function is not Riemann integrable.

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: If n is an even integer prove that Φ(2n) = 2Φ(n)

A: Given: n is an even integer.To prove: ϕ2n=2ϕn

Q: 2. Given a line that passes through the point (-2,1,3) and is parallel to the line joining A(1,5,3)…

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Q: If (ar + br) = 2n %3! n+1 and k-1 8. E(ak - br) = n+2 - , then ar %3D k-1 k-1

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Q: Let S(2) = ) for |z| <1. 1 n=1 zn Evaluate f(10) (0). You may leave factorials in your answer…

A: In the question we are given a complex function f(z) in terms of infinite series, and we have to…

Q: Show that z / [(z-1)(z-3)] = -3 Σ [(z-1)n / 2n+2] - [1 / (2(z-1))] for 0 < |z - 1| < 2.

A: by using techniques of partial fractions and series expansion of 1/(1-x)

Q: (-1)* x* k! k! k=0 k=0 2k а) 2 (k!)* k=0 b) 1 (-1)*. (2k)! k 2k k=0 d) 0 e) None of these.

A: Expansion of Exponential function

Q: 28. Show {Xn, n ≥ 1} are independent if o(X₁,..., Xn-1) || σ(Xn) are independent for each n ≥ 2.

A: Given::- Show (Xn, n ≥ 1} are independent if o (X1,..., Xn-1) || σ (Xn) are independent for each n ≥…

Q: 6. Let f(n)= o(n)/n. (a) Show that if p is prime, then f(p) = f(p). (b) Find all n such that f(n) =…

A: Introduction: Euler's Totient function is a mathematical multiplicative function that counts the…

Q: A container contains 35 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly…

A: There are total (35+5)=40 non-red tokens and 3 red tokens in the container.  And total (35+5+3)=43…

Q: 100 Compute Σ (-1) k=u

A: Given ∑k=0∞-12k

Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: n+1 Ei.2¹=n2"+2 +2 i=1 For all integer n >= 0

A:

Q: 2. (a) Show that k!n-k-1 = for n>0 and ka nonnegative integer. 5. e-nxxk dx

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Q: Wi <² -- = K (K+1)(2K+1) G

A: As you post multiple questions according to guidelines, I only solve the first one. Please post…

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: 50 Evaluate Σ (5 + 2i) i=1

A: Solution is done in step 2

Q: Q.4) Find the value of Sigma (d=sum of student id + 22, and c=12) 4c(d + x)(7c* +2cz +9z*) ((d +x)*…

A: As the student ID is not provided, I assume the sum of the digits are 10 (say). Then d=10+22=32 and…

Q: Compute ΣΣ k=0 n=0 (-1)n+knk 5nk!

A:

Q: Solve for the moment generating function of X if for all natural number n, X ∼ U {1,2,…,n}.

A: Given that, X ∼ U {1,2,…,n} , we need to solve for the moment generating function of X if for all…

Q: Prove that for n ≥ 1: 1k² = n(n + 1)(2n + 1).

A: Given that  to prove for n≥1, ∑k=1nk2=16n(n+1)(2n+1)

Q: *) Let p(x) = 8x – 7 and q(x) = /x – 7. Find u(x) given that q(x) = p(u(x)). - и(x) —

A: see 2nd step

Q: (d) If each fn has fewer than M discontinuities (where M ∈ N is fixed), thenf has fewer than M…

A: To Show: If each fn has fewer than M discontinuities (where M ∈ N is fixed), then f has fewer than M…

Q: 6. Show that (x, y) = 1/2 (||x + y||² − ||t||² — ||ý||²) -

A: Given; &lt;x,y&gt;=(llx+yll2-llxll2-llyll2)/2.

Q: 6. Show that when 0 < |z – 1| < 2, Σ (z-1)(z-3) 00 --Σ n=0 = (z-1)" 2n+2 1 2(z - 1)*

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Q: Find the singular values σ₁ ≥ σ2 of 01 = 02 -1 -27 A = -2 1 -2 2 9.

A: Find the singular values σ₁ ≥ σ₂ of the matrix A: Singular values of a matrix are the square roots…

Q: Show that Σ dn o(n) n for every positive integer n.

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Q: Consider k : D –→C k(2) = %3D (1 – z Show k is 1-to-1 and find k(D).

A: Consider a function is k : D→ℂ                                                     kz=z1-z2 To show:…