Real Analysis Answers

The function f defined by :

f(x) = -| x-3| , x ∈ R has a local maxima.

True or false with full explanation

The set { x∈ R : x≤ 5} is not a closed set in R.

True or false

Test the following series for convergence

2.3/(4^2.5^2)+ 4.5/6^2.7^2) + 6.7/ (8^2.9^2) ..+.......

Find the maclaurin series expansion of f(x)= 1/1+x hence or otherwise find the expansion of 1/1+x^2 by integrating both sides of your result, find the series expansion of arctanx, by also putting x=1 find the expression π/4 hence show that π=3.142

Show that the circumference of the ellipse (𝑥 ²/ 𝑎² )+( 𝑦 ²/ 𝑏²) = 1 is given by 2𝜋𝑎 [1- ∑^∞_𝑛=1 ( (2𝑛−1)!!/ (2𝑛)!! ) ²( 𝑒 ^2𝑛 /2𝑛−1 ) ]. Here 𝑒 = √(1 −( 𝑏²/ 𝑎² )), 𝑎 > 𝑏 is the eccentricity. Length of graph 𝑦 = 𝑓(𝑥) can be found by ∫ √(1 + ( 𝑑𝑦 /𝑑𝑥) ² )𝑑x.

Prove ∫^∞₀ 𝑒 ^{−𝑥 2} 𝑥 ² 𝑑𝑥 = √𝜋/ 4

Option 1: Company ABC’s selling stock is ₱ 1,500.00 per share that will have a dividend of ₱ 200.00 per year. The stock can be sold after two years at ₱2,000.00 and the market requires a rate of return of 15%. The stock can be sold after two years at ₱2,000.00 and the market requires a rate of return of 7%. Determine the stock yield ratio? *

ROUND OFF, NUMBERS ONLY, DO NOT PUT COMMA, PERIOD, ZERO OR PERCENTAGE SIGN.

Let f be a non- negative measurable function. Then there is a sequence <pi n> of simple function with pi n+1 ≥ pi n, such that f=lim pi n at each point of x.If f is defined on a sigma-finite measure space.we may choose the function pi n so that each vanishes outside a set of a finite measure.

prove that A set of measure zero need not be null set.

prove that A measurable set is a null set if and only if every measurable subset of it has gamma-measure zero.