Fix k, n E N with n > 2 and k

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

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Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

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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: Let 2a4 + y* + 23 = 4 Then dz (1, 1) = 8/3 -8/3 none of the listed here 1

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Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: A water tank has the shape of the solid formed when the region bounded by y = and the y-axis for 0 <…

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Q: 2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

A: Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns.  There is a relationship…

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

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Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

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Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: (3 = |x-"x) (W Z UE) (NWA) (0 UA) (NWE) (0 > 3A) O (3 = x - x)(W UA) (NWA) (0 x = ¹x) (W > UE)…

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Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: Prove & (2n) = $ &(n) if n is odd. ( 2. p(n) if n is even,

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: (1– (C+D)), provided (C+D) 4PQ. (28) Substituting (25) and (26) into (28), we get the condition…

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