Compute Lim Inf & Lim Sup of Sequence

[tracedinair]

Homework Statement

Given the sequence, 1/2 + 1/3 + 1/(2^(2)) + 1/(3^(2)) + 1/(2^(3)) + 1/(3^(3)) + ..., Describe the terms of the sequence and use it to compute the lim inf (a_n+1)/(a_n); lim sup (a_n+1)/(a_n); lim inf (a_n)^(1/n); lim sup (a_n)^(1/n).

Homework Equations

The Attempt at a Solution

First, I found the formula for the sequence, which is $$\Sigma$$ i=1 to infinity of [1/(2^(i)) + 1/(3^(i))].

I wrote out some terms of the sequence, but I'm having a hard time pulling out a subsequence to compute the ratio and root tests.

 

[Mark44]

First off, 1/2 + 1/3 + 1/(2^(2)) + 1/(3^(2)) + 1/(2^(3)) + 1/(3^(3)) + ... is an infinite sum (AKA infinite series), not a sequence. Every infinite sum has a sequence associated with it, the sequence of partial sums, {S_n}, which is defined this way:
$$S_n~=~\sum_{i = 1}^n a_n$$

Is a_n defined for your infinite sum? If not, it would be more convenient to define the general term in your series as a_n = 1/2ⁿ + 1/3ⁿ. If a_n is defined in this way or can be defined this way, the ratio a^{n + 1}/aⁿ is fairly easy to calculate, and comes out to (3ⁿ⁺¹ + 2ⁿ⁺¹)/(6(3ⁿ + 2ⁿ)). You should be able to work with this to get your lim sup and lim inf values

On the other hand, if a_n has a different formula for the even terms and the odd terms, it's going to be more difficult to calculate the aforementioned ratio.

 

[tracedinair]

The second part of the problem says I should consider both cases, positive and negative, when computing the lim inf and lim sup. I've never seen this type of problem before and I'm totally sure what it is asking me to do.

 

[Mark44]

You didn't include that information in your problem description. What do you mean, both cases, positive and negative?

 

[tracedinair]

Yeah, both cases positive and negative.

 

[Mark44]

Again, what do you mean by the positive and negative cases?

 

[tracedinair]

From my notes: After writing down a formula for {a_n}, describe {a_n} when n is even (n=2k) and when n is odd (2k-1).

 

[Mark44]

That information would have answered my questions in post #2. This doesn't have anything to do with positive and negative, but rather, odd and even terms.

So finally, we're getting somewhere.

Here are some more questions.
What do the a_2k terms of this series look like?
What do the a_{2k - 1} terms look like?

If these are too hard, what do a₂, a₄, a₆, and so on look like? Can you generalize this?

What do a₁, a₃, a₅, and so on look like? Can you generalize this?

 

[tracedinair]

How are you getting the lim inf and lim sup vales out of (3n+1 + 2n+1)/(6(3n + 2n))?

 

[Mark44]

I'm not. Look at post #8 again, and answer the questions I've asked.