frac{x - 1}{x + 1} + frac{1}{2} cdot frac{x^2 | vedclass

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(A) ધારો કે $S = \sum_{n=1}^{\infty} \frac{1}{n} \left( \frac{x^n - 1}{(x + 1)^n} \right) = \sum_{n=1}^{\infty} \frac{1}{n} \left( \left( \frac{x}{x +

Vedclass · Accepted Answer

$\log_e x$

Vedclass · Answer

(A) ધારો કે $S = \sum_{n=1}^{\infty} \frac{1}{n} \left( \frac{x^n - 1}{(x + 1)^n} \right) = \sum_{n=1}^{\infty} \frac{1}{n} \left( \left( \frac{x}{x + 1} \right)^n - \left( \frac{1}{x + 1} \right)^n \right)$.લોગરીધમિક શ્રેણીના વિસ્તરણ $\ln(1 - y) = - \sum_{n=1}^{\infty} \frac{y^n}{n}$ નો ઉપયોગ કરતા:$S = - \ln\left(1 - \frac{x}{x + 1}\right) - \left( - \ln\left(1 - \frac{1}{x + 1}\right) \right)$.$S = - \ln\left(\frac{1}{x + 1}\right) + \ln\left(\frac{x}{x + 1}\right)$.$S = \ln(x + 1) + \ln\left(\frac{x}{x + 1}\right) = \ln\left((x + 1) \cdot \frac{x}{x + 1}\right) = \ln x$.

$\frac{x - 1}{x + 1} + \frac{1}{2} \cdot \frac{x^2 - 1}{(x + 1)^2} + \frac{1}{3} \cdot \frac{x^3 - 1}{(x + 1)^3} + \dots \infty = $

Similar Questions

જો $P = 1 + \frac{1}{2 \times 2} + \frac{1}{3 \times 2^{2}} + \dots$ અને $Q = \frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \dots$ હોય, તો

જો $|x| < 1$ અને $y = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots$ હોય,તો $x$ ની કિંમત શું થાય?

$\log_{10}\left(\frac{n}{n-1}\right)$ ના વિસ્તરણમાં $n^{-r}$ નો સહગુણક શું છે?

જો $S = \sum\limits_{n = 0}^\infty \frac{(\log x)^{2n}}{(2n)!}$ હોય,તો $S$ =

$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{3 \cdot 4 \cdot 5} + \frac{1}{5 \cdot 6 \cdot 7} + \dots \infty = $

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