And what are the two numbers you think they converge to?Unassuming said:They both converge to two different numbers.
Unassuming said:1 and -1? I am beginning to doubt myself.
No, neither of them converges to 1 or -1! I have no idea how you could come to that conclusion.Unassuming said:1 and -1? I am beginning to doubt myself.
This notation is asking for a proof or demonstration that the series (-1)^n + (1/n) diverges. In other words, it is asking for evidence that the terms of this series do not approach a finite limit as n approaches infinity.
A series is an infinite sum of terms. It is represented by the notation ∑ (sigma) and can be written in the form ∑(an) = a1 + a2 + a3 + ... + an + ... , where an is each term in the series and n is the index or position of the term in the series.
A series diverges if the sum of its terms does not approach a finite limit as the number of terms increases. In other words, the series does not have a defined sum and the terms do not approach a specific value as n approaches infinity.
There are several tests that can be used to determine if a series diverges, such as the divergence test, comparison test, limit comparison test, and integral test. In this specific case, we can use the divergence test to show that the series (-1)^n + (1/n) diverges.
Yes, the proof is as follows:
By the divergence test, a series ∑(an) diverges if lim n→∞ an ≠ 0.
In this case, an = (-1)^n + (1/n).
As n approaches infinity, the terms alternate between 1 and 0, so the limit does not exist and is not equal to 0. Therefore, the series (-1)^n + (1/n) diverges.