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Hi everybody, so today we are going to be doing a step -by -step problem that's five parts that deals with infinite series.
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To begin, we are given this problem right here.
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A of n equals 6 divided by n squared plus 3n.
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And to start off, we want to know the partial fraction decomposition.
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So to start off with this, we are going to go ahead and just break this down as best as we can using partial fractions.
00:36
So to start off, we're going to draw an n out.
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And when we do that, we can break it down into aught, divide by n, plus a1, divide by n plus 3.
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And that's the partial fractions decomposition part of it.
01:01
From here, we'll want to solve for both a -ought and a -1.
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We'll start by simply multiplying to simplify everything down, so we'll get 6 equals a -a -a -t times n plus 3, plus a -1 -n.
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From there, you end up with a -a -a -t equals 2, and a -1 equals negative 2.
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So then you'll just go ahead and plug it back in to the original form.
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Formula.
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And once you do that, you'll get a of n equals 2 divided by n plus 2 divided by n plus 3.
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Our next problem we want to go ahead and write this out as an infinite series to the first five terms.
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So as you can see, i went ahead and wrote it down below...