If {an} (a sequence) is decreasing and an > 0 for all n, then {an} is convergent. True/False?
True.
When considering a SEQUENCE, such as {an} = 1/n^2, it will always be greater than 0, yet by an infinitely small amount as n approaches infinity. This is why we say it converges at 0, although it never actually reaches it.
{an} = 1/n the SEQUENCE will behave the same way. However, it is important to distinguish between series and sequences. The harmonic SERIES 1/n is the sum of every nth term, and will approach infinity.
False.
A decreasing sequence that is bounded below converges, but in this case, it is not specified whether the sequence {an} is bounded below or not. So, we cannot conclude whether {an} is convergent or not based solely on it being a decreasing sequence with an > 0 for all n.
To determine whether the sequence {an} is convergent given that it is decreasing and an > 0 for all n, we first need to understand what it means for a sequence to be convergent.
A sequence is said to be convergent if there exists a real number L such that for any positive tolerance (or error) ε, we can find a positive integer N such that |an - L| < ε for all n > N.
In other words, for a convergent sequence, as we go further along in the sequence, the terms get closer and closer to a particular value L.
Now, let's look at the given statement: "If {an} is decreasing and an > 0 for all n, then {an} is convergent."
This statement is false. The fact that the sequence {an} is decreasing and an > 0 for all n does not guarantee that the sequence is convergent.
Consider the sequence {an} = {1, 1/2, 1/3, 1/4, ...}. This sequence is clearly decreasing, and all the terms are positive. However, as we go further along in the sequence, the terms get arbitrarily close to zero but never actually reach a specific value. Therefore, the sequence {an} is not convergent.
In summary, the statement is false. Just because a sequence is decreasing and all terms are positive does not necessarily mean the sequence is convergent.
nope. consider {an} = 1/n
The harmonic series diverges.