Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 156776 by mnjuly1970 last updated on 15/Oct/21

$$$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}(nln\left(\frac{2n+1}{2n-1}\right)-1)=?$$$$$$

Commented by CAIMAN last updated on 15/Oct/21

1/2-1/2 ln2

Answered by qaz last updated on 15/Oct/21

$$\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}(\mathrm{nln}\frac{2\mathrm{n}+1}{2\mathrm{n}-1}-1)$$$$=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}(\mathrm{nln}(2\mathrm{n}+1)-\mathrm{nln}(2\mathrm{n}-1)-1)$$$$=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}{\int }_0^1(\frac{\mathrm{n}}{\mathrm{x}+2\mathrm{n}}-\frac{\mathrm{n}}{\mathrm{x}-2\mathrm{n}}-1)\mathrm{dx}$$$$={\int }_0^1\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}(\frac{\mathrm{n}}{\mathrm{x}+2\mathrm{n}}-\frac{\mathrm{n}}{\mathrm{x}-2\mathrm{n}}-1)\mathrm{dx}$$$$=\frac{1}{4}{\int }_0^1{\mathrm{x}}^2\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{1}{{\mathrm{n}}^2-{\left(\frac{\mathrm{x}}{2}\right)}^2}\mathrm{dx}$$$$=\frac{1}{4}{\int }_0^1{\mathrm{x}}^2\frac{1-\pi \left(\frac{\mathrm{x}}{2}\right)\cot \frac{\pi \mathrm{x}}{2}}{2{\left(\frac{\mathrm{x}}{2}\right)}^2}\mathrm{dx}$$$$=\frac{1}{4}{\int }_0^1(2-\pi \mathrm{xcot}\frac{\pi \mathrm{x}}{2})\mathrm{dx}$$$$=\frac{1}{4}(2-2\ln 2)$$$$=\frac{1}{2}(1-\ln 2)$$

Commented by mnjuly1970 last updated on 15/Oct/21

$$bravosirqaz$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com