Determining convergence of a sum

In summary, the conversation discusses a strategy for showing that the series a_n= |sin n| / n diverges. The idea is to create a subsequence with elements greater than a certain value, and then compare it to the divergent series 1/n. However, there is uncertainty about how to choose the terms for the subsequence. The conversation suggests a simpler approach, by proving that for every pair of numbers n, n+1, either |sin n| or |sin n+1| is greater than a certain value. This can then be used to show that a series b_k = a_2k-1 + a_2k, for all k in N, is greater than a divergent series.
  • #1
pk1234
11
0
I'd really appreciate some help with a sum of:

a_n= |sin n| / n

All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges.
However, I have no idea how to go about this. I don't know how to show, that there really is 'enough' of >epsilon>0 terms. I can't think of a pattern by which to choose only those n, for which a_n > epsilon>0, and I don't really know whether that's possible. If I knew that | sin n| > 0.1 for all n=4k -2 or something like that, it would probably be easy, but since that seems impossible, I don't know what to do.
 
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  • #2
pk1234 said:
I'd really appreciate some help with a sum of:

a_n= |sin n| / n

All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges.
However, I have no idea how to go about this. I don't know how to show, that there really is 'enough' of >epsilon>0 terms. I can't think of a pattern by which to choose only those n, for which a_n > epsilon>0, and I don't really know whether that's possible. If I knew that | sin n| > 0.1 for all n=4k -2 or something like that, it would probably be easy, but since that seems impossible, I don't know what to do.

No, that won't work. Try to think more basic. Suppose |sin n|<1/10. Can you show |sin(n+1)|>1/10? That would mean for every pair of numbers n, n+1 then either |sin n| or |sin n+1|>1/10. What could you do with that?
 
  • #3
Thanks.

If I prove that, should I then take a series of:
b_k = a_2k-1 + a_2k, and say that for all k element N, b_k >= (1/10) * (1/2k) = 1/20k, which diverges?
 
  • #4
pk1234 said:
Thanks.

If I prove that, should I then take a series of:
b_k = a_2k-1 + a_2k, and say that for all k element N, b_k >= (1/10) * (1/2k) = 1/20k, which diverges?

Sure that's the idea. And proving my statement can be done pretty crudely. Just draw a graph of the sine function and it should be easy to convince yourself.
 

Related to Determining convergence of a sum

1. What is the definition of convergence in mathematics?

Convergence in mathematics refers to the behavior of a sequence or series as the number of terms increases. It is the idea that as you continue to add terms to a sequence or series, the values will eventually approach a specific limit or value.

2. How is convergence of a sum determined?

The convergence of a sum is determined by examining the behavior of the terms in the series. If the terms get smaller and smaller as the series progresses, and eventually approach zero, then the series is said to converge. If the terms do not approach zero, then the series is said to diverge.

3. What is the difference between absolute convergence and conditional convergence?

Absolute convergence refers to a series where the absolute value of the terms converges, meaning that the series will converge regardless of the order in which the terms are added. Conditional convergence, on the other hand, refers to a series where the terms themselves converge, but the series only converges when the terms are added in a specific order.

4. Can a series converge even if the terms do not approach zero?

Yes, a series can converge even if the terms do not approach zero. This is known as the alternating series test, which states that if the terms alternate in sign and decrease in magnitude, then the series will converge.

5. What is the importance of determining the convergence of a sum?

Determining the convergence of a sum is important in many areas of mathematics and science. It allows us to understand the behavior of a series and make predictions about its future values. It is also crucial in analysis, as it helps us determine the convergence of integrals and other important mathematical operations.

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