Determine if the series converges absolutely, converges conditionally, or diverges.Justify
Q
n=1
((-1)^n)/(9n^1/4 +1)
To determine if the series converges absolutely, converges conditionally, or diverges, we will consider two separate series: one with terms of the positive values and one with terms of the negative values.
Let's consider the series with terms of positive values:
∑ (1/(9n^(1/4) + 1))
Using the limit comparison test, we can compare this series to the harmonic series:
∑ (1/n)
Taking the limit as n approaches infinity of the ratio of the two series gives us:
lim (n→∞) (1/(9n^(1/4) + 1))/(1/n)
= lim (n→∞) (n/(9n^(1/4) + 1))
As n approaches infinity, the fraction n/(9n^(1/4) + 1) approaches 1, which is a nonzero finite number.
Therefore, the series with terms of positive values converges if and only if the harmonic series converges. Since the harmonic series diverges, the series with terms of positive values also diverges.
Now let's consider the series with terms of negative values:
∑ (-1)^n/(9n^(1/4) + 1)
We can rewrite this series as two separate series: ∑ (1/(9n^(1/4) + 1)) and ∑ (-1/(9n^(1/4) + 1)).
We have already determined that the series ∑ (1/(9n^(1/4) + 1)) diverges.
As for the series ∑ (-1/(9n^(1/4) + 1)), we can use the alternating series test to determine if it converges.
The terms of the series are decreasing in magnitude, and the limit as n approaches infinity of the absolute value of the terms is 0. Therefore, the series ∑ (-1/(9n^(1/4) + 1)) converges.
Since one series within the original series diverges and the other converges, we conclude that the series ∑ ((-1)^n/(9n^(1/4) + 1)) converges conditionally.