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Question

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

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Solution

Let, Sn=.6+.66+.666+...tonterms=6[0.1+0.11+0.111+...tonterms]=69[0.9+0.99+0.999+...tonterms]=69[(1110)+(11102)+(11103)+...tonterms]=23[(1+1+...nterms)110(1+110+1102+...nterms)]=23⎢ ⎢ ⎢ ⎢n110⎜ ⎜ ⎜ ⎜1(110)n1110⎟ ⎟ ⎟ ⎟⎥ ⎥ ⎥ ⎥=23n230×109(110n)=23n227(110n)

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