How to tell if this series converges by alternating series test

In summary, the problem involves finding bn, which is the absolute value of an, for the given series. After removing the -1 from the expression, bn can be simplified to n/10n. The task is to prove that the limit of bn as n approaches infinity is equal to zero, and to show that bn+1 is less than or equal to bn. The use of derivatives to determine if the function is decreasing or increasing is considered, but the presence of 10n adds difficulty in applying this method. Assistance is requested in solving this problem.
  • #1
vande060
186
0

Homework Statement



(∞, n=1) ∑ ((-1)n)/(10n)


Homework Equations





The Attempt at a Solution



(∞, n=1) ∑ ((-1)nn)/(10n)

first I have to find bn, which is absolute value of an, so I can now forget the -1 of the expression, and I am left with:

bn = n/10n

I now have to show that the lim ans n approaches infinity is zero, which I am not too sure about, and I have to show that bn+1 < or equal to bn

I know I can take the derivative to see if the function is decreasing or not, but the 10n is really messing me up
 
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  • #2
, I'm not sure how to apply the definition of decreasing/increasing in this case. Any help would be greatly appreciated.
 

1. What is the alternating series test?

The alternating series test is a method for determining whether an infinite series converges or diverges. It specifically applies to series where the terms alternate in sign, such as 1 - 1/2 + 1/3 - 1/4 + ...

2. How do I know if I should use the alternating series test?

The alternating series test can be used if the series alternates in sign and the absolute value of each term decreases as the series goes on. It is also useful for series with terms that approach zero but do not necessarily decrease in value.

3. What is the process for using the alternating series test to determine convergence?

To use the alternating series test, you must first check that the alternating series satisfies the conditions for convergence (alternating signs and decreasing absolute value). Then, you can apply the test which states that if the limit of the absolute value of the terms approaches zero as n goes to infinity, then the series converges.

4. Can the alternating series test determine divergence?

Yes, the alternating series test can determine divergence. If the alternating series does not satisfy the conditions for convergence, then it must diverge.

5. Are there any other methods for determining convergence or divergence of series?

Yes, there are several other tests and methods for determining convergence or divergence of series, such as the ratio test, integral test, and comparison test. Each method has its own set of conditions and limitations, so it is important to carefully choose the appropriate test for each series.

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