Find the sum of the following series up to n | vedclass

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(A) Let $S_n = 0.6 + 0.66 + 0.666 + \dots$ to $n$ terms.$S_n = 6 [0.1 + 0.11 + 0.111 + \dots 	ext{ to } n 	ext{ terms}]$$S_n = \frac{6}{9} [0.9 + 0.

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

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(A) Let $S_n = 0.6 + 0.66 + 0.666 + \dots$ to $n$ terms.$S_n = 6 [0.1 + 0.11 + 0.111 + \dots 	ext{ to } n 	ext{ terms}]$$S_n = \frac{6}{9} [0.9 + 0.99 + 0.999 + \dots 	ext{ to } n 	ext{ terms}]$$S_n = \frac{2}{3} [(1 - 0.1) + (1 - 0.01) + (1 - 0.001) + \dots 	ext{ to } n 	ext{ terms}]$$S_n = \frac{2}{3} [n - (0.1 + 0.01 + 0.001 + \dots 	ext{ to } n 	ext{ terms})]$The term in the bracket is a geometric progression with $a = 0.1$ and $r = 0.1$.$S_n = \frac{2}{3} [n - \frac{0.1(1 - (0.1)^n)}{1 - 0.1}]$$S_n = \frac{2}{3} [n - \frac{0.1(1 - 10^{-n})}{0.9}]$$S_n = \frac{2}{3} [n - \frac{1}{9}(1 - 10^{-n})]$$S_n = \frac{2}{3} n - \frac{2}{27}(1 - 10^{-n})$

Find the sum of the following series up to $n$ terms:
$0.6 + 0.66 + 0.666 + \dots$

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Find the sum of the following series up to $n$ terms:$0.6 + 0.66 + 0.666 + \dots$