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JH
Numerade Educator

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Problem 32 Medium Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$$ \displaystyle \sum_{n = 1}^{\infty} [(-0.2)^n + (0.06)^{n - 1}] $$

Answer

The series converges to $\frac{7}{3}$

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Video Transcript

let's determine whether this Siri's over here is conversion or diversion. If it commercials, then we'LL go and find the song. So before we look at the Siri's if you look inside the Siri's, you see a sum here. So let's just look at the two Siri's individually and see whether these converge the's air. Both geometric for this one. Over here, we see our is negative. Zero coin, too. Here, ours Cyril point zero six. And we know that a geometric series Comm urges. If that's Lou, value are is less than one. Since both of these are ours will satisfy that both Siri's will converge So by both. I'm referring to these two. And now we're just using a result that says that if these two theories individually converge, then you can go ahead and take the summation decimation outside and distributed to both terms. So here we have to separate geometric series. We know they both converge, really, because we looked at the our values over here. And, you know, the sum of a geometric series is one not one first term of the Siri's over one minus R. So in this problem, for the first time, if you plug in and equals one, that's your first term and then one minus R. And now for the second first term, plug in and equals one. You get this number here, This decimals with zero power, which is one and then one minus your are such is using this formula two times. Now, let's just go to the next page to simplify this last term down here so I can write. This is and then I could simplify this denominator. Six over five, then plus one hundred over ninety four. So here and this is after cancelling those fives there. And then we find that we have well, two, five, three over to a two. And this is our final answer. This is the sum of the conversion series.