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03/01/2022 12:56 pm
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Find the value of \(\frac{3}{2}x^2\)+\(\frac{5}{3}x^3\) + \(\frac{7}{4}x^4\)+.......∞ if 0 < x < 1.
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03/01/2022 1:06 pm
Let S = \(\sum_{r=1}^{∞}\)\(\frac{2r+1}{r+1}x^{r+1}\)
= 2\(\sum_{r=1}^{∞}x^{r+1}\) - \(\sum_{r=1}^{∞}\frac{x^{r+1}}{r+1}\)
= \(2[x^2+x^3+.... ∞]\)-\(\frac{x^2}{2}+\frac{x^3}{3} + \frac{x^4}{4}+..... ∞\)
= \(2[\frac{x^2}{1-x}]\)-\(x + \frac{x^2}{2}+...... -x\)
ln(1-x) = -x - \(\frac{x^2}{2} -\frac{x^3}{3}-\frac{x^4}{4} - ...... ∞\)
= \(\frac{2x^2}{1-x}\)+x + ln(1-x)