Prove that sin (n^2) + sin (n^3) is not a convergent

Thread starter Kummer
Start date Nov 9, 2007
Tags

Convergent Sin

In summary, convergence in mathematics refers to a sequence or series of numbers approaching a specific value as the number of terms increases. A convergent sequence approaches a specific value, while a divergent sequence does not have a limit and can either approach infinity or oscillate between values. To prove a sequence is not convergent, we must show it either approaches infinity or oscillates between values. The sequence sin(n^2) + sin(n^3) does not approach a specific value as n increases, making it not convergent. Its limit does not exist, as the values oscillate between -2 and 2. Understanding the behavior of sequences and their limits is important in solving complex mathematical problems.

Nov 9, 2007

Kummer

Prove that [tex]\sin (n^2) + \sin (n^3)[/tex] is not a convergent sequence.

Physics news on Phys.org

Nov 11, 2007

quasar987

Science Advisor

Homework Helper

Gold Member

Unpleasant!

Nov 11, 2007

mathwonk

Science Advisor

Homework Helper

how about just sin(n). that's already interesting. as opposed to sin(pi n) for example.

Nov 12, 2007

Kummer

mathwonk said:

how about just sin(n).

I believe that is easier. Consider n=3,31,314,3141,31415,...

Related to Prove that sin (n^2) + sin (n^3) is not a convergent

What is the definition of convergence in mathematics?

In mathematics, convergence refers to the idea that a sequence or series of numbers approaches a specific value as the number of terms increases.

What is the difference between a convergent and a divergent sequence?

A convergent sequence approaches a specific value as the number of terms increases, while a divergent sequence does not have a specific limit and can either approach infinity or oscillate between values.

How can we prove that a sequence is not convergent?

To prove that a sequence is not convergent, we need to show that it either approaches infinity or oscillates between values. In this case, we can show that the sequence sin(n^2) + sin(n^3) does not approach a specific value as the number of terms increases, therefore it is not convergent.

What is the limit of the sequence sin(n^2) + sin(n^3) as n approaches infinity?

The limit of this sequence does not exist, as the values oscillate between -2 and 2 as n increases. Therefore, the sequence is not convergent.

What is the significance of proving that sin(n^2) + sin(n^3) is not a convergent?

Proving that a sequence is not convergent is important in understanding the behavior of the sequence and its limit. It also helps in identifying patterns and relationships between different sequences, which can be useful in solving complex mathematical problems.