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

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

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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: 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: 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: 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

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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: 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: 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:…