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Jac8897
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Homework Statement
Hi
I want to see if you guys could check my work an tell me if i did it correct and also explain me how to cancel "look at the graph"
Homework Equations
The Attempt at a Solution
Last edited:
NastyAccident said:Yes.
When in doubt, set a term an equal to what is being summed. In this case, an = 2^n/n!.
Now, get the next term in the series so an+1. In this case, an+1 = 2^(n+1)/(n+1)!.
Then, plug in an and an+1 into [tex]|\frac{a_{n+1}}{a_{n}}|[/tex]
Now, the n! cancels because n! is equivalent to n*(n-1)(n-2)(n-3)(n-4). Whereas (n+1)!, is equivalent to (n+1)(n)(n-1)(n-2)(n-3)(n-4).
So, as you can see, the terms to the right of (n+1) are canceled out.
NastyAccident
The ratio test is a method used to determine the convergence or divergence of a series. It compares the ratio of consecutive terms in the series to a limit, and if the limit is less than 1, the series is convergent. It is often used as a more efficient alternative to other convergence tests, such as the comparison test or the integral test.
To apply the ratio test, you must first find the limit of the ratio of consecutive terms in the series. This can be done by simplifying the expression and using algebraic techniques, or by applying L'Hopital's rule. Once you have found the limit, you must compare it to 1 to determine the convergence or divergence of the series. If the limit is less than 1, the series is convergent, and if the limit is greater than 1, the series is divergent.
The ratio test can be used for any series, as long as the terms in the series are non-negative. If the terms are negative, the test may give incorrect results. Additionally, the series must also have non-zero terms, meaning that each term in the series must have a non-zero value.
If the limit of the ratio is equal to 1, the ratio test is inconclusive. This means that the test cannot determine the convergence or divergence of the series and other tests or methods must be used to determine the behavior of the series.
No, the ratio test is used only to determine the convergence or divergence of a series. To find the sum of a series, you must use other methods such as the geometric series formula or the partial sum formula. The ratio test only provides information about the convergence or divergence of a series, not its actual sum.