- #1
karush
Gold Member
MHB
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$$\sum_{k=1}^{\infty}\left(\frac{k}{k+1}\right)^{2k}$$
$\textsf{how do you use absolute converge test on this?}$
$\textsf{how do you use absolute converge test on this?}$
karush said:$\textsf{Find the radius / interval of convergence }\\$
\begin{align}
\displaystyle f(x)&=2k(x-1)^k
\end{align}
$\textsf{thot would ask another questions here since new stuff?}$
MarkFL said:If we are given:
\(\displaystyle f(x)=\sum_{k=0}^{\infty}\left(2k(x-1)^k\right)\)
Then we compute the radius of convergence as follow:
\(\displaystyle |x-1|<\lim_{k\to\infty}\left|\frac{2k}{2(k+1)}\right|=1\)
The absolute converge test is a method used to determine if a series converges or diverges. It involves taking the absolute value of each term in the series and then testing if the resulting series converges or diverges.
The absolute converge test should be used when the terms in a series alternate in sign or when the series contains only positive terms. In these cases, the absolute value of the terms can help determine the convergence or divergence of the series.
The absolute converge test works by taking the absolute value of each term in a series and then comparing the resulting series to known series that either converge or diverge. If the resulting series is similar to a convergent series, then the original series also converges. If the resulting series is similar to a divergent series, then the original series also diverges.
The absolute converge test can be useful in determining the convergence or divergence of a series when other tests, such as the ratio or root tests, are inconclusive. It can also be used to simplify a series by removing alternating signs or negative terms.
Yes, the absolute converge test is not always conclusive and may give incorrect results for certain types of series. It should always be used in conjunction with other tests to confirm the convergence or divergence of a series.