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Of Integrals and Summations

\[ \int_{a}^{b} f(x) \ \mathrm{d}x = (b-a) \sum_{n=1}^{\infty} \sum_{k=1}^{2^n - 1} \dfrac{(-1)^{k+1}}{2^{n}} f \left( a+ \left(\frac{b-a}{2^n}\right) k \right) \]

Prove the identity above, given that the function \(f\) has a bounded variation on \([a,b]\).


This is a part of the set Formidable Series and Integrals

Note by Ishan Singh
1 year, 2 months ago

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