How many integers between 14 and 3,452 can be divided by 42 with a remainder of zero?
1 answer:
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Answer:
I.
CORRECT.
II.
CORRECT.
III.
CORRECT.
IV.
CORRECT.
V.
INCORRECT.
VI.
CORRECT
Step-by-step explanation:
To understand let us restate the comparison test in simple terms.
Comparison test :
Given and
such that
, then
1. If converges then
converges as well.
2. If diverges then
diverges as well.
Now to give you a more intuitive idea of what is going on, think about it like this. When the series on top converges it is like an "upper bound" for what you have on the bottom, therefore what you have on the bottom has to converge as well.
Similarly if what you have on the bottom explotes, then what you have on top will explote as well.
That's how I like to think about that intuitively.
Now, using those results let us examine the statements.
I.
Since the infinite sum of 1/n diverges in fact the infinite sum of ln(n)/n does not converge.
Therefore, CORRECT.
II.
Since the infinite sum of is in fact convergent then
converges as well using the comparison theorem. Therefore
CORRECT.
III.
Once again does converge so what you have on the bottom converges as well. Therefore
CORRECT.
IV.
Once again converges therefore since it is on top what is on the bottom converges as well. Therefore.
CORRECT.
V.
Now the fact that diverges does not necessarily imply that what you have on the bottom diverges. Therefore
INCORRECT.
VI.
That is correct as well since what you have on top converges therefore what you have on the bottom converges as well.