Question

Use an appropriate method to evalualte the limit, if it exists, of each of the following sequences {ai}. If a sequence is divergent, explain why.

Answer 1

\[\frac{ 5^i-2^{3i+1} }{ 7^{i-2}+8^i }\]

Answer 2

Try writing this sequence as a combination of other geometric sequences.

Answer 3

What do you mean? Like manipulating it?

Answer 4

Yes. Notice how each term in the numerator and denominator is some base to a power of the index?

First thing you can do is break up the sequence:
\[\frac{5^i-2^{3i+1}}{7^{i-2}+8^i}=\frac{5^i}{7^{i-2}+8^i}-\frac{2^{3i+1}}{7^{i-2}+8^i}\]
The next thing you could do is compare to another sequence that you know converges/diverges.

For example, you know that \(\dfrac{1}{8^i}\) is a convergent geometric sequence, right? Well, you know that \(\dfrac{1}{7^i+8^i}<\dfrac{1}{8^i}\), which means the left sequence also converges.

Answer 5

I don't understand how you know \[\frac{ 1 }{ 7^{i}+8^i }<\frac{ 1 }{ 8^i }\]

Answer 6

wait never mind okay lol .

Answer 7

So for this given sequence, you know that
\[\frac{5^i}{7^{i-2}+8^i}<\frac{5^i}{8^i}\]
converges.

Finding the limit is the tricky part, but shouldn't be too hard if you know L'Hopital's rule.

Answer 8

Actually, disregard that thing I mentioned about L'Hopital's rule. You know that geometric sequences converge to 0 if it's a convergent sequence.

Answer 9

So it is just 0

Answer 10

Yes, it looks that way.

Answer 11

At least for the first "sub-sequence" we've written. We haven't considered the one with \(2^{3i+1}\) yet.

Answer 12

the top exponent is throwing me off, how would you figure out that?

Answer 13

Hmm, maybe
\[-\frac{2^{3i+1}}{7^{i-2}+8^i}=-2\left(\frac{2^{3i}}{\dfrac{7^i}{49}+8^i}\right)=-2\left(\frac{8^{i}}{\dfrac{7^i}{49}+8^i}\right)\]
Then you dom some factoring:
\[-2\left(\frac{1}{\dfrac{7^i}{49(8^i)}+1}\right)=-2\color{red}{\left(\frac{49}{\dfrac{7^i}{8^i}+49}\right)}\]
The \(\dfrac{7^i}{8^i}\) approaches 0, so the red term becomes 1.

Answer 14

Which means that this entire sequence approaches -2.

Answer 15

Where did the 7/49 come from? D: