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JH
Numerade Educator

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Problem 37 Medium Difficulty

The meaning of the decimal representation of a number $ 0.d_1d_2d_3. . . .$ (where the digit $ d_i $ is one of the numbers 0, 1, 2, . . . , 9) is that
$ 0.d_1d_2d_3d_4 . . . = \frac {d_1}{10} + \frac {d_2}{10^2} + \frac {d_3}{10^3} + \frac {d_4}{10^4} + . . . $
Show that this series always converges.

Answer

$\sum_{n=1}^{\infty} \frac{d_{n}}{10^{n}}$ will always converge by the Comparison Test.

Discussion

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Video Transcript

Let's show that this some on the right hand side always converges. So here the d ies these were the digits, and they're all numbers between zero and I. So this means that the one over tens d to over ten square and so on if I just keep adding and going in this direction forever, that this is less than or equal to if I just go ahead and replace all of the DEA's with nine. So this is just by using this fact up here for D one, then we did it for D two and so on, and we keep doing it for all these. Now we see that the Siri's on the right hand side, this's geometric, and we see that there are what are we multiplying by each time? Just won over ten. So this will converge and we even know what the sum is. You take the first term of the series, and then he just divide by one minus R. So in this case, the first term is nine over ten, and then our was won over ten, so one minus one over ten. So we have nine over ten over nine over ten, and that equals one. So on the other hand, we know that D I over ten for any number. I is always bigger than or equal to zero since d eyes bigger than or equal to zero. This the reason I'm pointing that out is because if we want to use the comparison test, we need to make sure that we're on ly dealing with positive terms. It's part of the hypothesis for the hero. So we just shown that our Siri's the one on the left hand side, which I'll Circle and Blue. We wanted to know whether this converged. We bounded in above by a larger Siri's that was Geum. Measure it that converges toe one. So we know that our Siri's also converges So d one over ten D to over ten square and so on disc convergence by the comparison test. And that's our final answer