Let S = { /12 - 2/2 | n € N₁m = N} (considered as a subset of R). N, n m (a) Prove that sup S = 1. (b) Is 1 € s? (c) Does S have an infimum? If so, what is it?

Q: 1/sqrt(1) + 1/sqrt(2) + . . . + 1/sqrt(n) = sqrt (n) for n=1,2,3

A: We have to prove  11+12+13+...+1n≥n For n=1 11=1 For n=2   Suppose for n=k the statement is true…

Q: p(x) = a + ax + a,x² + ... + 10x?

A:

Q: 13. Use induction on k to EIR", then X₁1 | x,t... Xklls x + prove that if +||XK||

A:

Q: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

Q: Euler's Tution function: (n) is the number of positive integers less than n that are relatively…

A:

Q: (b) For any n ≥ 1, the factorial function, denoted by n!, is the product of all the positive…

A:

Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

A:

Q: x₁ + x₂ + +x. Prove that -x₁)(1+x₂)... (1+x) ≤1+S+ ... S² S³ + 2! 3! +...+ uS - n!

A:

Q: Define the term infimum. Consider the set Prove that inf A = 0. n A = {- m² : n, m € Z¹, m≤n}.

A:

Q: If n > 1 is a composite number, then σ(n) > n+sqrt(n).

A: ⚫ For a given positive integer n, σ(n) denotes the sum of positive divisors of n. For example ,…

Q: Compute (a) E-ok · k! = (n+ 1)! – 1. (b) D (") = (). k=0 TIL

A: The problem can be solved using concepts of permutation and combination.