WEBVTT
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okay, if we rewrite this, this is one
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plus X squared plus X to the fourth. Plus
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that that that And then we also have these two
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x to execute etcetera terms. We can factor out
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a two x from all those guys and write it
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as two acts times one plus x squared plus X
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to the fourth. Plus that that. Okay,
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so remember that one plus x plus X squared plus
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dot, dot dot is one over one minus x
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. Here. We're doing the same thing, except
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instead of using acts were using X squared. So
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this is going to be one over one minus x
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squared, and then, similarly, over here,
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this is too X divided by one minus X squared
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. So we have two x plus one over one
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minus x squared. Okay, So this is the
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explicit formula that we have and the interval of convergence
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for this whole sum is going to be the same
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thing as the interval convergence for this guy here,
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right. We can rewrite this whole thing as two
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x plus one quantity multiplied by this infinite sum.
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So this infinite sum is the only thing that we
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have to really worry about here. Right? And
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this infinite sum is just X to the two n
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from any Kools zero to infinity. Okay, so
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you can use the ratio test. Figure out what
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this is in case you forgot what this radius of
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convergence is here. Using the ratio test, we
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would get absolute value of X to the two n
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plus one divided by X to the two end.
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So this is going to turn out to just be
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X squared. And to get that to be less
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than one acts would be between minus one and one
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. The radius of convergence here is one and our
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values of accer between minus one and one. So
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now we just need to check the end points.
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So we plug in X equals minus one. Here
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we get one plus one plus one plus one.
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Is that all? Certainly be divergent if you plug
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in one here. Same thing. One plus one
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plus one plus one divergent. So our interval of
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convergence is minus one. The one not including them
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. So open in trouble. Here