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  • Find the sum of the following series up to n terms: .6 +. 66 +. 666+…

21(ii)   Find the sum of the following series up to n terms: 

               .6 +. 66 +. 666+…

Answers (1)

best_answer

Sum of  0.6 +0. 66 + 0. 666+….................

It can be written  as 

S_n=0.6+0.66+0.666+.......................... to n terms

S_n=6[0.1+0.11+0.111+0.1111+................]

S_n=\frac{6}{9}[0.9+0.99+0.999+0.9999+................]

S_n=\frac{6}{9}[(1-\frac{1}{10})+(1-\frac{1}{10^2})+(1-\frac{1}{10^3})+(1-\frac{1}{10^4})+................]

S_n=\frac{2}{3}[(1+1+1.....................n\, terms)-\frac{1}{10}(1+\frac{1}{10}+\frac{1}{10^2}+...................n \, terms)]

S_n=\frac{2}{3}[n-\frac{\frac{1}{10}(\frac{1}{10}^n-1)}{\frac{1}{10}-1}]

S_n=\frac{2n}{3}-\frac{2}{30}[\frac{10(1-10^-^n)}{9}]

S_n=\frac{2n}{3}-\frac{2}{27}(1-10^-^n)

Posted by

seema garhwal

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