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Convergence Test
If \(\underset{n\to\infty}{\lim}a_n\neq0\) then \( \sum a_n \) is guaranteed to diverge
If \(\underset{n\to\infty}{\lim}a_n=0\) then the test is inconclusive use another test.
Convergence Test
\(\displaystyle\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}\, \text{if}\, |x|\)<\(1\) Otherwise if \(|x|\geq1\) then the series in question will diverge.
Convergence Test
Convergence Test
Convergence Test
Convergence Test
Convergence Test
Convergence Test
Convergence Test
\(\displaystyle\sum_{n=k}^{\infty}x^n=\frac{x^k}{1-x}\, \text{if}\, |x|\)<\(1\) Otherwise if \(|x|\geq1\) then the series in question will diverge.
Convergence Test
Convergence Test
Convergence Test
Convergence Test
Convergence Test
Convergence Test
To determine the expect error for \(S_n\) for a given alternating series \(S=\sum(-1)^na_n\) requires investigating the inequality \(|S-S_n|<a_{n+1}\).
In other words, the expected error corresponds to the value of the \(a_{n+1}\)
We can bound the sum of alternating series (that passes the alternating series test) by noting the following dervitions.
\(\begin{align*} |S-S_n| &\leq a_{n+1}\\ -a_{n+1}&\leq S-S_n \leq a_{n+1}\\ S_n-a_{n+1}&\leq S\leq S_n+a_{n+1} \end{align*}\)
To determine the (minimum) number of terms \(n\) that guarantees that an approximation \(S_n\) for a given alternating series \(S=\sum(-1)^na_n\) is within the a given toleration \(\rho\) requires solving the inequality \(a_{n+1}<\rho\) in terms of \(n\).
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