Let an 20 for all n and suppose Σan converges. Prove that √an ΣΥ n converges. Hints: One way to do this is to consider the inequality (√an-1)² ≥ 0.

Q: Let = e²i/n be an n-th root of unity. Show that (k(1-n) Resck (27²1) n for all 0 <k<n-1. =

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Q: a) If {anao and {bn}a_o are sequences of positive real numbers such that an << bn, then bn tan <<…

A: As per the question we are asked to examine and provide proofs or counterexamples for a series of…

Q: Use |S – SN| <bN+1 to find the smallest value of N such that Sy approximates the value of the sum S…

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Q: Find all the values of t so that 9eBt > 0. Ot≥0 00 21/1/20 Ot< 1/1/2 9 -00 < t < 00

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Q: Alan will throw a six-sided fair die repeatedly until he obtains a 2. Bob will throw the same die…

A: Given information: Alan will throw a six-sided fair die repeatedly until he obtains a 2. Bob will…

Q: 1.31. Let p be a prime and let q be a prime that divides p - 1. (a) Let a € F and let b = a(P-¹)/4.…

A: To prove that either b = 1 or b has order q in the field , where p is a prime and q is a prime that…

Q: Let (an) be a sequence of positive terms, and suppose that (an) → L. Show that (√) → √ī.

A: Proof is given below Explanation:Step 1:Step 2: Step 3: Step 4:

Q: Find all the values of t so that 8et > 0. O-∞ 21/1/2 8 Otzo

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Q: Find a closed formula for S(n, n − 2), for all n ≥ 2.

A: Find a closed formula for S(n, n − 2), for all n ≥ 2.

Q: For 1 ≤p ≤ ∞, find the closer of Coo in P. (Hint: It may be practical to separate into two cases, 1…

A: We have to find the closer of C00 in lp

Q: B3.3.4.1f) (1) Use limit laws and SL. 1 - SL. 8 for sequences, to evaluate the following limits: cos…

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

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Q: Find k such that P(-k &lt; Z &lt; k) = .6970  it might help to sketch the situation

A: To value of k can be determine as:By using Z-table,

Q: 3.2.6 Theorem If X (x) then a < lim(x) b. is a convergent sequence and if a S X5bfor all n e N,

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Q: Let E = . Determine the smallest choice of N > 0 such that N. |n(n+1)|

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Q: Determine whether an = limit if it converges. In (5n+2) In (2n-3) converges or diverges and…

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Q: Let a 1 and define an+1 = V4an -I for n2 1. Show that the sequence {an} converges and find the…

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Q: 2. (Arithmetic Mean-Geometric Mean Inequality) Let n E N and {r;}"-1 be nonnegative numbers, prove…

A: Use mathematical induction and check if the statement is true for n=1. The left hand side of the…

Q: Let X ∼ Binomial(n, p). Using Chebyshev inequality, find an upper bound on P(X ≥ αn), where p < α <…

A: We know from Chebyshev's inequality: P(X≥μ+a)≤σ2σ2+a2, where a&gt;0 Here, X~Bin(n,p). Then, μ=np and…

Q: How do you know that sup R is a lower bound for S? How do you know that it is greater than or equal…

A: Given: S⊂ℝ is a non-empty set which is bounded below. A set R is defined as R=b∈ℝ|b is a lower bound…

Q: Let an 6n 2n + 1 (a) Determine whether {a} is convergent. O convergent divergent = (b) Determine…

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Q: Exercise 10.2.4. In view of Equation (10.2.2), one might suppose that the following is true: (Σ²).…

A: Given equality: ∑i=0nxi·∑i=0nyi=∑i=0nxiyi   a) Verify the equality is always true. If not, give an…

Q: 5. Show that the infinite sequence a, = Vn2 + 1 – yn² – 1 (n 1) converges by showing that it is…

A: Given sequence is  an= √(n^2+1) - √(n^2-1)

Q: Use Bernoulli's inequality to prove that n + √)" = 1+ > √n for all n EN.

A: From Bernoulli inequality we know that  if x&gt;-1 then : 1+xn≥1+nx for n≤0 or n≥1. Therefore for…