Real Analysis Answers

Convergence test for "\\displaystyle\\sum_{n=1}^\\infty \\frac{sin(n)}{n}".

create your own real life situation where exponential function is applied

Show that ∫¹₀ (sin(1/x))/xⁿ 𝑑𝑥 ; 𝑥 > 0 convergence absolutely, if 𝑛 < 1.

Find the convergence of the following series,

1. ∑^∞ _𝑛=1(√𝑛+1−√𝑛)/ 𝑛  
2. ∑^∞ _𝑛=1 cos(1/n)
3. ∑^∞ _𝑛=1 (sin n)/n
4. ∑^∞ _𝑛=1 ((𝑛!) ²)/((2𝑛)!)
5. ∑^∞ _𝑛=1 (n/n+1)n²

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)

Check the convergence of the sequence defined by 𝑢𝑛+1 = 1 2 (𝑢𝑛 + 𝑎 𝑢𝑛 ) , 𝑎 > 0. Note that this is the sequence associated with finding the square root of a number 𝑎 > 0 by the Newton’s method

If 𝑓 is a continuous function on [0,1], show that lim 𝑛→∞[0 to 1] ∫ 𝑛𝑓(𝑥)/(1+n^2.𝑥^2) 𝑑𝑥 = 𝜋/2. 𝑓(0)

If 𝑓 is a continuous function on [0,1], show that lim𝑛→∞ ∫ 𝑛𝑓(𝑥) 1+𝑢2𝑥 2 1 0 𝑑𝑥 = 𝜋 2 𝑓(0). 

Check the convergence of the sequence defined by 𝑢𝑛+1 = 1 2 (𝑢𝑛 + 𝑎 𝑢𝑛 ) , 𝑎 > 0. Note that this is the sequence associated with finding the square root of a number 𝑎 > 0 by the Newton’s method.

A rectangular field is to be enclosed on four sides with a fence. Fencing costs $4 per foot for two opposite sides, and $7 per foot for the other two sides. Find the dimensions of the field of area 740 ft 2 that would be the cheapest to enclose.

A. 36 ft @ $4 by 20.6 ft @ $7

B. 20.6 ft @ $4 by 36 ft @ $7

C. 47.6 ft @ $4 by 15.5 ft @ $7

D. 15.5 ft @ $4 by 47.6 ft @ $7