Question and Answers Forum

All Questions      Topic List

UNKNOWN Questions

Previous in All Question      Next in All Question      

Previous in UNKNOWN      Next in UNKNOWN      

Question Number 56424 by gunawan last updated on 16/Mar/19

The sum of first 10 terms of the series (x+ (1/x))^2 + (x^2 + (1/x^2 ))^2 + (x^3 + (1/x^3 ))^2 +... is

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{10}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\left({x}+\:\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} \:+\:\left({x}^{\mathrm{2}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} \:+\:\left({x}^{\mathrm{3}} +\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)^{\mathrm{2}} +...\:\mathrm{is} \\ $$

Answered by mr W last updated on 16/Mar/19

a_n =(x^n +(1/x^n ))^2 =x^(2n) +2+(1/x^(2n) ) Σ_(n=1) ^(10) a_n =Σ_(n=1) ^(10) x^(2n) +10×2+Σ_(n=1) ^(10) (1/x^(2n) ) =((x^2 (1+x^(20) ))/(1+x^2 ))+20+(((1/x^2 )(1+(1/x^(20) )))/(1+(1/x^2 ))) =20+(((1+x^(20) )^2 )/(x^(18) (1+x^2 )))

$${a}_{{n}} =\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{\mathrm{2}} ={x}^{\mathrm{2}{n}} +\mathrm{2}+\frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} } \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{a}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{x}^{\mathrm{2}{n}} +\mathrm{10}×\mathrm{2}+\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} } \\ $$$$=\frac{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{20}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }+\mathrm{20}+\frac{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{20}} }\right)}{\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \\ $$$$=\mathrm{20}+\frac{\left(\mathrm{1}+{x}^{\mathrm{20}} \right)^{\mathrm{2}} }{{x}^{\mathrm{18}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com