Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 144199 by qaz last updated on 23/Jun/21

$$\mathrm{Prove}\mathrm{that}$$$$1+\frac{1^2}{6+\frac{3^2}{6+\frac{5^2}{6+\frac{7^2}{6+\frac{9^2}{6+...}}}}}=\frac{2}{1+\frac{1\times 2}{1+\frac{2\times 3}{1+\frac{3\times 4}{1+\frac{4\times 5}{1+\frac{5\times 6}{1+...}}}}}}$$

Answered by Dwaipayan Shikari last updated on 23/Jun/21

$$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+..=1+(-\frac{1}{3})+(-\frac{1}{3})(-\frac{3}{5})+...$$$$=\frac{1}{1+\frac{1/3}{1-\frac{1}{3}+\frac{3/5}{1-\frac{3}{5}+\frac{5/7}{1-\frac{5}{7}+\frac{7/9}{1-\frac{7}{9}..}}}}}=\frac{1}{1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+..}}}}}$$$${tan}^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+...$$$$=\frac{x}{1+\frac{x^2/3}{1-x^2/3+\frac{3x^2/5}{1-3x^2/5+..}}}=\frac{x}{1+\frac{x^2}{3-x^2+\frac{3^2{x}^2}{5-3x^2+\frac{5^2{x}^2}{7-5x^2+..}}}}.$$$$...$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com