Mathquery

Uniform Convergence :        Uniform Convergence is a part of real analysis which is discussed in detail below. Definition :             Let (X,d) be a metric space and f be a function from X to R . Also for each n ∈ ℕ let fₙ : X ---> R . Then , the sequence of functions <fₙ> converges pointwise to the function f , if for each x ∈ X , the sequence of real numbers < fₙ(x) > converges to the real number f(x) .                   Therefore <fₙ(x) > converges pointwise to f if  lim   fₙ(x) = f(x)    ∀ x ∈ X .                              n-->∞ For Example :            Let < fₙ > be the sequence defined by fₙ  : R --> R such that fₙ(x) = x / n  ∀ x∈R , n∈N  . Show that the sequence converges pointwise to zero function . Solution :          Here , we have to show that the given sequence converges pointwise to the zero function i.e f(x) = 0 , x∈ R , then we must show that given ε > 0 ,we can find m∈ ℕ  such that 

Examples Related To Riemann Integral :  Example 1 :                                                                                                1                  Show that ∫ x⁴ dx = 1 / 5                                     0  Proof :                     Let us consider the partition P in the interval [0,1] as { 0,1/n, 2/n , 3/n .....n/n }.  Since  f(x) = x⁴ , so the supremum and infimum of the function in the interval is (i/n)⁴ and (i-1 / n) ⁴  .    Length of the interval = Δxᵢ = 1-0 / n = 1/n                             n                 n      So U( P,x⁴ ) = Σ Mᵢ Δxᵢ = Σ  (i/n) . 1/n                            i=1              i=1                                                  n                                      = 1/n⁵ Σ i⁴                                                 i = 1                              = 1/n⁵ [1⁴ + 2⁴ +.......+n⁴]            = (1/n⁵ ) n(n+1)(2n+1)(3n+1)(4n+1)/120    = 1/120  (1+1/n)(2+ 1/n)(3 +