Disputed Sequence

Algebra Level 1

What is the decimal representation of series below? ∑ n = 1 ∞ 9 1 0 n = 9 10 + 9 100 + 9 1000 + 9 10000 + … \sum_{n=1}^{\infty} \frac{9}{10^{n}} = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{9}{10000} + \ldots


The answer is 1.

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Guillermo Templado by Guillermo Templado , 46, Spain

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3 solutions

Mateus Gomes
Feb 14, 2016

∑ n = 1 ∞ 9 1 0 n = 9 10 + 9 100 + 9 1000 + 9 10000 + … ( G P ) = 9 10 1 − 1 10 = 1 \Large\sum_{n=1}^{\infty} \frac{9}{10^{n}} = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{9}{10000} + \ldots~~(\color{#D61F06}{GP})=\frac{\frac{9}{10}}{1-\frac{1}{10}}=\large\color{#302B94}{\boxed{\color{#D61F06}{\boxed{1}}}}

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nice answer.. (y)

Aswad Hariri Mangalaeng - 5 years, 4 months ago
Sravanth C.
Feb 14, 2016

∑ n = 1 ∞ 9 1 0 n = 9 10 + 9 100 + 9 1000 + 9 10000 + … = 0.9 + 0.09 + 0.009 + 0.0009 … = 0.99999999 ⋯ = 1 \sum_{n=1}^{\infty} \frac{9}{10^{n}} = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{9}{10000} + \ldots\\=0.9+0.09+0.009+0.0009\dots\\=0.99999999\dots=\boxed 1

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Actually, @Mateus Gomes 's solution is the appropriate way to solve the problem. You haven't shown why 0.999.. =1.

Mehul Arora - 5 years, 3 months ago

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I know, I was tired of explaining it time and again that I got bored and left it as it is . . .

Sravanth C. - 5 years, 3 months ago

Let x =9/10+9/100+9/1000+....
then, x=9/10+1/10(9/10+9/100+9/1000....
x=9/10+1(x)/10.
Solving for x, we get,
x=1.



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