Real Analysis Answers

Show whether the following functions are uniformly continuous on the given domain.

1. F(x)=x^3 on [-1,1]

2. F(x)= 2x/2x-1 on [1, infinity]

3. F(x)= sinx/x on (0,1)

4. F(x)= 1/x on (0,1)

If a>0 then show that Lin(1/1+na)=0

27. On December 31, 2020, the partnership of Abe, Bravo, Charlie and Delfin decided to liquidate with the following account balances. Cash NCA Liab. Bravo Charlie Abe Bravo Charlie Delfin Loan Loan (10%) (20%) (45%) (25%) 20,000 180,000 100,000 10,000 5,000 15,000 25,000 15,000 30,000 Assuming that Bravo and Charlie are limited partners and Abe, Bravo, Charlie and Delfin are solvent up to P10,000; P15,000; P10,000 and P5,000 respectively; determine the cash settlement given to Abe if the NCAS were sold for P100,000 (round to nearest peso).

For the function tan^(−1) 𝑥  find the infinite Taylor series at 𝑎 = 0, the radius of convergence, range of convergence, derivative, integral and the product with itself.  

Given that ∑ 𝑢𝑘 ∞ 𝑘=1 converges with 𝑢𝑘 > 0, prove that ∑ √𝑢𝑘.𝑢𝑘+1 ∞ 𝑘=1 also converges. Show that the converse is also true if 𝑢𝑘 is monotonic.

Let 𝑥𝑘 be a Cauchy sequence and 𝑓 be a continuous function, show that 𝑓(𝑥𝑘) is also a Cauchy sequence. 

Check the convergence of the sequence defined by 𝑢𝑛+1 = (1 + 1/ 𝑢𝑛 ) , 𝑢1 > 0. Note that this is the sequence associated with the continued fraction expansion of the Golden ratio. 

Check the convergence of the sequence defined by 𝑢𝑛+1 = (1 + 1 𝑢𝑛 ) , 𝑢1 > 0. Note that this is the sequence associated with the continued fraction expansion of the Golden ratio. 

Check the convergence of the sequence defined by 𝑢𝑛+1 = 𝑎/ 1+𝑢𝑛 where 𝑎 > 0, 𝑢1 > 0.

A pendulum of length 𝑙 at an angle 2𝛼. Find the time period T of the pendulum. Also let 𝛼 → 0 and obtain the well-known formula 𝑇 = 2𝜋√ 𝑙 /𝑔 .