- #1

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Here's something I'm having difficulty seeing:

Suppose

[tex]u_n > 0[/tex] and

[tex]\frac{u_{n+1}}{u_n} \leq 1-\frac{2}{n} + \frac{1}{n^2}[/tex] if [tex]n \geq 2[/tex]

Show that [tex]\sum{u_n}[/tex] is convergent.

I'm not sure how to apply the ratio test to this.

It looks like I would just take the limit.

I get: [tex]lim_{n \rightarrow \infty} 1-\frac{2}{n} + \frac{1}{n^2} = 1[/tex]

I'm not sure if I'm correct, but I could see this two ways.

Since the above limit converges to 1, then the summation converges by the ratio test.

Or, since the limit converges to one, the summation may converge or diverge.

Is either statement correct? Am I on the right track?

Thanks for the help.

Bailey