Series Convergence: Ratio Test & Lim. n→∞

In summary, the ratio test is a method used to determine whether a series converges or diverges by taking the limit of the absolute value of the ratio of consecutive terms. If the limit is less than 1, the series converges, and if it is greater than 1 or infinite, the series diverges. It is also used to determine absolute convergence, but cannot determine conditional convergence. The ratio test is a more general test than other tests for series convergence, but may yield an inconclusive result for some series.
  • #1
Confusedalways
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I'm trying to determine if \(\displaystyle \sum_{n=1}^{\infty}\frac{{n}^{10}}{{2}^{n}}\) converges or diverges.

I did the ratio test but I'm left with determining \(\displaystyle \lim_{{n}\to{\infty}}\frac{(n+1)^{10}}{2n^{10}} \)

Any suggestions??
 
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  • #2
I would write the limit as:

\(\displaystyle L=\frac{1}{2}\lim_{n\to\infty}\frac{(n+1)^{10}}{n^{10}}=\frac{1}{2}\left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^{10}\)

Can you proceed?
 
  • #3
Yes thanks!
 

FAQ: Series Convergence: Ratio Test & Lim. n→∞

What is the ratio test for series convergence?

The ratio test is a method used to determine whether a series converges or diverges. It involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges.

What is the limit as n approaches infinity in the ratio test?

The limit as n approaches infinity in the ratio test represents the behavior of the series as the number of terms increases. It is an important factor in determining whether the series converges or diverges. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges.

How do you apply the ratio test to a series?

To apply the ratio test to a series, you first find the absolute value of the ratio of consecutive terms in the series. Then, take the limit as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges. If the limit is exactly 1, the test is inconclusive and another method must be used to determine convergence or divergence.

Can the ratio test determine absolute or conditional convergence?

The ratio test can only determine absolute convergence. Absolute convergence means that the series converges regardless of the order in which the terms are added. The test cannot determine conditional convergence, which means that the series converges only when the terms are added in a specific order.

What is the relationship between the ratio test and other tests for series convergence?

The ratio test is a more general test than the comparison and limit comparison tests. It can be used to determine convergence or divergence of series that cannot be evaluated using these other tests. However, for some series, the ratio test may yield an inconclusive result, in which case another test must be used to determine convergence or divergence.

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