If > 0 and (an+1/ converges to p and p<1, prove converges to zero.

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- September 26th 2007, 08:17 AMcowgirl123divisor convergence?
If > 0 and (an+1/ converges to p and p<1, prove converges to zero.

- September 26th 2007, 08:40 AMJhevon
- September 26th 2007, 10:17 AMThePerfectHacker
This means, . It must be the case that because the sequence of ratios is a sequence of non-negative terms. So the limit is non-negative. I leave the case to you to prove. I will assume , i.e. . The important step is to notice that there most exists so that . This means for we have by convergence, for . Thus, for .

This means,

.

In general,

for .

This tells us that the sequence is bounded by for all .

Thus,

.

Now, .

Because and it is a geometric sequence.

Q.E.D.