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Ahat [919]
3 years ago
11

. Evaluate the series 1 + 0.1 + 0.01 + . . .

Mathematics
2 answers:
EastWind [94]3 years ago
6 0

We can employ a simple repeated decimal trick:

x=1.111\ldots

0.1x=0.111\ldots

\implies x-0.1x=1\implies0.9x=1\implies x=\dfrac1{0.9}=\dfrac{10}9

###

Alternatively, we can compute the partial sum of the series.

\displaystyle S_n=\sum_{k=0}^n\dfrac1{10^k}

S_n=1+0.1+0.01+\cdots+\dfrac1{10^n}

0.1S_n=0.1+0.01+0.001+\cdots+\dfrac1{10^{n+1}}

\implies S_n-0.1S_n=0.9S_n=1-\dfrac1{10^{n+1}}

\implies S_n=\dfrac{10}9-\dfrac9{10^n}

As n\to\infty, the second term vanishes and we're left with \dfrac{10}9. Notice that this is really just a more formal version of the earlier trick.

amm18123 years ago
5 0

Answer:

1.11

Step-by-step explanation:

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Assuming you meant 2/5 because 2.5 would mean there are more roses in the vase than there are flowers in the vase, heres how to solve this.

Understanding

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