Give an example of an increasing sequence that is bounded between 5 and 7. Does your sequence converge or diverge? Note that you need to prove that your example satisfies the requirements.

Question

Give an example of an increasing sequence that is bounded between 5 and 7. Does your sequence converge or diverge? Note that you need to prove that your example satisfies the requirements.

Answer 1

Let take sequence a_n = 2.5(1 + 1/n)ⁿ;

We know that this sequence monotonic increasing and converges to 2.5e = 6.8 < 7

and min a_n = a₁ = 5. So, 5 ≤ a_n < 7 

Answer 2

Hi Charles T.,

i forget whether a sequence is defined as a series of terms that you may add together, or as the ongoing cumulative sum of those terms. In either event, "bounded" means that the absolute value of successive terms of the sequence stays within the bound limits. I'll let you figure out the mechanics, but a convergent sequence might be (as ongoing cumulative sums) 5.5, 6.5, 6.0, 6.25, 6.125, .... and a divergent (which is anything that doesn't converge!) sequence might be: 5.5, 6.5, 5.5, 6.5, 5.5 ....

this would also be diverging, as individual terms: 5.5 - 6.5 + 5.5 - 6.5 + 5.5 .... You can't have a comparable converging sequence expressed in that way, since the successive term signs are either all the same, or alternate, and each term would have to satisfy 5<|x_i|<7 .

-- Cheers, --Mr. d.