for the series is absolutely convergent and convergent
the series from 1 to infinity of (x^3)/(5^x)
i did the ration test and get absolutely convergent and convergent
is this correct?
To determine if a series is absolutely convergent or convergent, you have correctly applied the Ratio Test, which is a common method used to analyze the convergence of series.
For the given series:
∑[(x^3)/(5^x)], as x ranges from 1 to infinity,
Let's apply the Ratio Test to check for convergence.
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 converges absolutely. If the limit is equal to 1 or the limit does not exist, further investigation is needed to determine the convergence or divergence.
In this case, the general term is a[n] = (x^3) / (5^n).
Applying the Ratio Test:
lim(n→∞) |(a[n+1] / a[n])| = lim(n→∞) |[(x^3) / (5^(n+1))] / [(x^3) / (5^n)]|
= lim(n→∞) |(x^3) / 5|
Since x^3 and 5 are constants, we can treat them as such when we take the limit. Thus, we have:
lim(n→∞) |(x^3) / 5|
Now, let's analyze this limit:
- If |(x^3) / 5| < 1, then the series converges absolutely.
- If |(x^3) / 5| = 1, then the test is inconclusive and further investigation is needed.
- If |(x^3) / 5| > 1, then the series diverges.
Therefore, based on the Ratio Test, if |(x^3) / 5| < 1, the series ∑[(x^3)/(5^x)] from 1 to infinity converges absolutely. If |(x^3) / 5| ≥ 1, further methods or tests need to be applied to determine the convergence or divergence.
Please note that this analysis assumes x is a fixed constant and does not depend on n. If x depends on n, additional steps may be required to analyze the series.