Find the sum of the following series: (i) 0.6 + 0.66 + 0.666 + …. to n terms.

Home » Find the sum of the following series: (i) 0.6 + 0.66 + 0.666 + …. to n terms.

Find the sum of the following series:
(i) 0.6 + 0.66 + 0.666 + …. to n terms.

Find the sum of the following series:
(i) 0.6 + 0.66 + 0.666 + …. to n terms.

Solution:

(i) $0.6+0.66+0.666+\ldots$ to $n$ terms.
Let,s take 6 as a common term therefore we obtain,
$6(0.1+0.11+0.111+\ldots n$ terms $)$
Now multiplying and dividing by 9 we obtain,
$6 / 9[0.9+0.99+0.999+\ldots+n$ terms $]$
$6 / 9[9 / 10+9 / 100+9 / 1000+\ldots+n$ terms]
We can write this as
$\begin{array}{l} 6 / 9[(1-1 / 10)+(1-1 / 100)+(1-1 / 1000)+\ldots+\mathrm{n} \text { terms }] \\ 6 / 9\left[\mathrm{n}-\left\{1 / 10+1 / 10^{2}+1 / 10^{3}+\ldots+\mathrm{n} \text { terms }\right\}\right] \\ 6 / 9\left[\mathrm{n}-1 / 10\left\{1-(1 / 10)^{\mathrm{n}}\right\} /\{1-1 / 10\}\right] \\ 6 / 9\left[\mathrm{n}-1 / 9\left(1-1 / 10^{n}\right)\right] \end{array}$