MHB Which Test Should I Use: Ratio Test or Root Test in Series Convergence?

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To determine whether to use the ratio test or the root test for series convergence, consider the nature of the series' terms. The ratio test is suitable for series with terms that simplify well through division, such as those involving powers of n or factorials. Conversely, the root test is more effective for series where taking the root simplifies the expression, particularly when terms include "n" in the exponent. Understanding the structure of the series will guide the choice of test. Proper application of these tests can clarify whether a series converges or diverges.
aruwin
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Hello.
How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.
 

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aruwin said:
Hello.
How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.

If your series has terms that will likely simplify with division (such as powers of n or factorials), then the ratio test is appropriate.

If your series will likely simplify by taking a root (e.g. because it has "n" in the power) then the root test is appropriate.
 
Prove It said:
If your series has terms that will likely simplify with division (such as powers of n or factorials), then the ratio test is appropriate.

If your series will likely simplify by taking a root (e.g. because it has "n" in the power) then the root test is appropriate.

Thank you!
 
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