theorem1 let an and bn be squences of real numbers theorem 2 let an and bn and cn be squences of real numbers if an<bn<cn theorem 3 let an be squences of real numbers if an=L and L define...
Let {an} m-o and {bn}ņ=be any two sequences of real numbers, we define the following: N • For any real number L € R, we write an = L if and only if lim Lan = L. N-0 n=0 n=0 X • We write an = bn if and only if there is a real number L such that n=0 n=0 I and Σ. = L. Select all the correct sentences in the following list: X η (Α) Σ Σ...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
please do both question clearly,thank you! Consider the sequence of real numbers 13. 1 1 2' 1 2 + 2 + 1 2 + Show that this sequence is convergent and find its limit by first showing that the two sequences of alternate terms are monotonic and finding their limits. Prove that any sequence in hence we may suppose that each subsequence has a least term.) (Note that this result and the theorem on the convergence of bounded monotonic sequences...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
This assignment asks you to prove the following Proposition 1 Let {n} and {n} are two sequences of real numbers and L is a number such that (1.a) un → 0, and (1.b) V EN, -L Swn. We illustrate the proposition. To begin, one can check from the definition that 1/n 0. This fact, plus the arithinetic rules of convergence, generate a large family of sequences known to converge to 0. For example, 11n +7 1 11 +7 3n2 -...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Limits at Ifinity and Limits of Sequences 787 57. Business The cost function for a certain model of a personal digital assistant (PDA) is given by C I 3.50x + 45,750, where C is the cost (in dollars) and x is the number of PDAs produced. (a) Write a model for the average cost per unit produced (b) Find the average costs per unit when 100 and x 1000. (c) Determine the limit of the average cost function as x...
Please solve the exercise 3.20 . Thank you for your help ! ⠀ Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...