7. Let fn [a, b] → R be Riemann integrable and suppose that fn → funiformly on [a, b]. (a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any partition of [a.b]. Use the definition of the uniform convergence of fn to f to show that there exists an N € N so that S(f, P) ≤ S(ƒN, P) + €. (b) Prove that lim m. [*J₁ (2) "fu(2)de = $* f(2) dz. fn dx n→∞

7. Let fn [a, b] → R be Riemann integrable and suppose that fn → funiformly on [a, b]. (a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any partition of [a.b]. Use the definition of the uniform convergence of fn to f to show that there exists an N € N so that S(f, P) ≤ S(ƒN, P) + €. (b) Prove that lim m. [J₁ (2) "fu(2)de = $ f(2) dz. fn dx n→∞

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.1: Numerical Integration

Problem 21E

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