Is this correct?
determine if absolutely convergent and convergent
1. the series from n=0 to infinity of ((-1)^n)/n!
I said it was abs. conv, and therefore conv
2. the series from n=0 to infinity of
(-1)^n/(the square root of (n^2+n+1))
I said ratio test was inconclusive so not abs. conv. but conv. from the A.S.T.
3. the series from n=1 to infinity of (-1)^(n+1)/n^4
I wasn't sure how to do this one, I know not abs. conv. because the ratio test was one but how do you tell if conv?
To determine if a series is absolutely convergent and convergent, you can use various convergence tests such as the ratio test, comparison test, alternating series test, or the integral test.
1. For the series from n=0 to infinity of ((-1)^n)/n!:
To determine absolute convergence, we can use the ratio test. The ratio test states that if the limit of the absolute value of (a[n+1]/a[n]) as n approaches infinity is less than 1, then the series is absolutely convergent.
In this case, the terms of the series are ((-1)^n)/n!. If we take the absolute value of each term, we get (1/n!). Now, applying the ratio test:
lim(n->∞) |((-1)^(n+1)/(n+1)!)/((-1)^n/n!)|
= lim(n->∞) |((-1)^(n+1)/(n+1)!)*(n!)/(-1)^n|
= lim(n->∞) |-1/(n+1)|
= 0.
Since the limit is less than 1, the series is absolutely convergent.
Since the series is absolutely convergent, it is also convergent.
2. For the series from n=0 to infinity of (-1)^n/(√(n^2+n+1)):
For this series, you mentioned that the ratio test was inconclusive. Inconclusive means that the ratio test does not give a definitive answer. In such cases, you can try using other convergence tests.
In this case, we can use the alternating series test (AST) since the terms alternate in sign. The AST states that if a series (-1)^n*b[n] is such that b[n] is positive, decreasing, and approaches zero as n approaches infinity, then the series is convergent.
For our series, (-1)^n/(√(n^2+n+1)), we can see that the terms alternate in sign and the denominator (√(n^2+n+1)) is positive, decreasing, and approaches zero as n approaches infinity. Therefore, the series is convergent.
However, since the absolute value of the terms, |(-1)^n/(√(n^2+n+1))| = 1/(√(n^2+n+1)), does not approach zero, the series is not absolutely convergent.
3. For the series from n=1 to infinity of (-1)^(n+1)/n^4:
For this series, you correctly identified that it is not absolutely convergent because the ratio test yields a result of 1. To determine convergence, we can use another test, such as the alternating series test or the comparison test.
Using the alternating series test, since the terms alternate in sign, and the absolute value of the terms, |(-1)^(n+1)/n^4| = 1/n^4, is decreasing and approaches zero as n approaches infinity, the series is convergent.
Hence, the series from n=1 to infinity of (-1)^(n+1)/n^4 is convergent but not absolutely convergent.