Determine the convergence or divergence:
An=(-1)^n(n/(n+1))
A(n+1)= -1^(n+1) (n+1)/(n+2)
A(n+1)/An= -1* 1/n * 1/(n+2) which as n >> large, converges to zero.
I would reccomend you to open your book and spend a few minutes ;)
To determine the convergence or divergence of the sequence An = (-1)^n(n/(n+1)), we can use the Alternating Series Test.
The Alternating Series Test states that if a series An = (-1)^nBn is such that Bn+1 ≤ Bn for all n and lim Bn = 0, then the series converges.
Let's break down the given sequence:
An = (-1)^n(n/(n+1))
We can rewrite the sequence as:
An = (-1)^n * n/(n+1)
Now, let's observe the sequence term by term.
When n is odd, the numerator (-1)^n * n is negative, and the denominator (n+1) is positive. So, An is negative.
When n is even, the numerator (-1)^n * n is positive, and the denominator (n+1) is positive. So, An is positive.
Now, let's calculate the limit of the absolute value of the sequence:
lim |An| = lim |(-1)^n * n/(n+1)|
As n approaches infinity, the magnitude of n/(n+1) approaches 1. Therefore, the limit of the absolute value of An is:
lim |An| = lim |(-1)^n * 1| = 1
Since the limit of the absolute value of An is equal to 1, but not 0, the Alternating Series Test does not guarantee convergence or divergence in this case.
To further investigate, we can use the Ratio Test.
The Ratio Test states that if lim |An+1/An| = L, where L is a finite number, then the series converges if L < 1 and diverges if L > 1.
Let's apply the Ratio Test to the given sequence:
lim |(An+1)/(An)| = lim |((-1)^(n+1) * (n+1)/((n+1)+1)) / ((-1)^n * n/(n+1))|
Simplifying the expression above, we get:
lim |(-1)(n+1)/(n+2)| = 1
Since the limit of the absolute value of the ratio is equal to 1, the Ratio Test is inconclusive.
Therefore, we cannot definitively determine the convergence or divergence of the given sequence (-1)^n(n/(n+1)) using the Alternating Series Test or the Ratio Test.