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