Maths · Limits, continuity and differentiability
If =\lim _{n \rightarrow \infty}\left[2 x+4 x^{3}+\dots \dots+\right. \) (0<x<1)
If \( f(x)=\lim _{n \rightarrow \infty}\left[2 x+4 x^{3}+\dots \dots+\right. \) \( \left.2 n x^{2 n-1}\right](0<x<1) \) then \( \int f(x) d x \) is equal to
- A. \( -\sqrt{1-x^{2}}+ \) constant
- B. \( \frac{1}{\sqrt{1-x^{2}}}+ \) constant
- C. \( \frac{1}{x^{2}-1}+ \) constant
- D. \( \frac{1}{1-x^{2}}+ \) constant
Step-by-step solution
The series is ∑_{k=1}^∞ 2k x^{2k-1} = d/dx(∑_{k=1}^∞ x^{2k}) = d/dx(x^2/(1-x^2)) = 2x/(1-x^2)^2. Then ∫ f(x) dx = ∫ 2x/(1-x^2)^2 dx = 1/(1-x^2) + C.
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