Write out correct answer choice below and show all work (homework non graded)

Question

#18 if 0< b subscript n <a subscript n for n is greater than or equal to 1, which of the following must be true?

1. If limit as n approaches infinity a subscript n=0 then series Upper limit infinity; lower limit n=1 function b subscript n converges
2. series Upper limit infinity; lower limit n=1 function b subscript n converges then limit as n approaches infinity b subscript n=0
3. If series Upper limit infinity; lower limit n=1 function b subscript n diverges, then limit as n approaches infinity a subscript n =0 
4. If series Upper limit infinity; lower limit n=1 function a subscript n diverges, then series Upper limit infinity; lower limit n=1 function b subscript n diverges
5. If series Upper limit infinity; lower limit n=1 function b subscript n converges , then series Upper limit infinity; lower limit n=1 function a subscript n converges

Answer 1

To determine which statement must be true, let's analyze the given information:

We are given that for all positive integers n (greater than or equal to 1), 0 < b_n < a_n, and that the limit as n approaches infinity of a_n is 0.

Now, let's consider the options:

1. **If the limit as n approaches infinity of a_n is 0, then the series with the upper limit of infinity and the lower limit of 1 for the function b_n converges.**

This statement is not necessarily true. The limit of a_n being 0 doesn't directly imply that the series of b_n will converge. The behavior of b_n, especially in relation to its series, isn't defined by the limit of a_n alone.

2. **If the series with the upper limit of infinity and the lower limit of 1 for the function b_n converges, then the limit as n approaches infinity of b_n is 0.**

This statement is generally not true. Convergence of a series does not necessarily mean that the individual terms in the sequence (b_n) tend to 0 as n approaches infinity.

3. **If the series with the upper limit of infinity and the lower limit of 1 for the function b_n diverges, then the limit as n approaches infinity of a_n is 0.**

This statement is also not necessarily true. The behavior of a_n, especially in relation to its series, isn't defined by the divergence of b_n's series.

4. **If the series with the upper limit of infinity and the lower limit of 1 for the function a_n diverges, then the series with the upper limit of infinity and the lower limit of 1 for the function b_n diverges.**

This statement is generally true. If the series of a_n diverges, and if 0 < b_n < a_n for all n, then the series of b_n will also diverge because the terms of b_n are always less than the corresponding terms of a_n. In other words, b_n can't be smaller than a_n and still result in a convergent series when a_n itself diverges.

5. **If the series with the upper limit of infinity and the lower limit of 1 for the function b_n converges, then the series with the upper limit of infinity and the lower limit of 1 for the function a_n converges.**

This statement is not necessarily true. The convergence of b_n's series doesn't imply the convergence of a_n's series, as the individual terms of b_n being smaller than a_n doesn't guarantee this.

So, the statement that must be true in this case is **Option 4**:

**If the series with the upper limit of infinity and the lower limit of 1 for the function a_n diverges, then the series with the upper limit of infinity and the lower limit of 1 for the function b_n also diverges.**