29, [2] * Fix k, n є P. Show that n+k-1 a2k-1 where the sum ranges over all compositions (a,... ,...
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix a valuc q with r <<1. Use the definition of r to prove that there exists a valuc N such that for any k 2 N. (b) Prove that Σο, laNIqk-1 converges, where N is the value from part (a)....
2. Show that p-1 E(+1) = 2a +(*)(4+1) k=0 l=0 Hint: Fix an 0 <I<p – 1, Ask youself how many k satisfy k2 = 1 mod p.
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Show that for all large positive integers n the sum 1/(n+1) + 1/(n+2) + 1/(n+3) + ... + 1/(2n) is approximately equal to 0.693. I am trying to solve this problem by setting the sigma summation from k = n + k to 2n of 1/j to try to make a harmonic sum but is not working. I let j be n + k so it matches the harmonic sum definition of 1/k
Topology (a) For each subset A of NV0), define eA є loo such that the k-th component of eA is 1 if k є A and 0 otherwise. Define B-(Bde (eA; 1/2) : A N\ {0)). Recall I. (i) If AメB are subsets of N \ {0), find the value of doc (eA, eB). (ii) Show that B is a collection of disjoint open balls in 100. iii) By quoting relevant results, justify whether or not the collection B is...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp 3. Let f: RP-R (a) If f(x)-Ax + b,...
QUESTION 6 (2 pts). Exercise 2.3.2 Suppose A є Mn,n(F) and that λ is an eigenvalue of A. Show that, for any choice of vector norm on Fn, we have lAll-A, where |All is the associated matrix norm of A. QUESTION 6 (2 pts). Exercise 2.3.2 Suppose A є Mn,n(F) and that λ is an eigenvalue of A. Show that, for any choice of vector norm on Fn, we have lAll-A, where |All is the associated matrix norm of A.
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n) (1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n)
1. a) Show that: , equals: AGpuC")_ACC"= Hem , assuming AHxn is constant over these ranges. b) Calculate AGxn of the reaction at 500K given AHPx = -92.22 kJ/mol and AG n=-32.9 kJ/mol at 298 K. N2 (g) + 3 H2 (9) → 2 NH3(g)
1. Fix n and k. How many positive integer solutions are there to x1 + + xk = n where xi i for all i?