∫abf(x) dx=(b−a)∑n=1∞∑k=12n−1(−1)k+12nf(a+(b−a2n)k) \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) ∫abf(x) dx=(b−a)n=1∑∞k=1∑2n−12n(−1)k+1f(a+(2nb−a)k)
Prove the identity above, given that the function fff has a bounded variation on [a,b][a,b][a,b].
This is a part of the set Formidable Series and Integrals
Note by Ishan Singh 4 years, 6 months ago
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
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