The sequence b(n)= npicos(npi) coverges or diverges?

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In summary, the sequence b(n)= npicos(npi) can converge or diverge depending on the value of n. If n is an integer, it will converge to 0, but if n is a half-integer, it will diverge. The limit of the sequence also depends on the value of n, with integer values resulting in a limit of 0 and half-integer values resulting in no limit and divergence. To determine if a sequence is convergent or divergent, we look at whether its terms approach a finite limit as n approaches infinity. A convergent sequence has a finite limit, while a divergent sequence does not. Other factors that can affect the convergence or divergence of a sequence include its form, the values
  • #1
Simkate
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The sequence b(n)= npicos(npi) coverges or diverges? if it converges what is the limit?

My Work

When n is even cosnpi= o and b(n)=o (therefore when n is oven it converges)
but
when n is odd cosnpi=-1 and b(n)= - infinity( thereofre when n is even it diverges)

is this correct?
 
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  • #2
cos(npi) is never zero. I think you are thing about sin(npi).

cos(npi) simply osscillates between 1 and -1.
 

1. Does the sequence b(n)= npicos(npi) converge or diverge?

The answer to this question depends on the value of n. If n is an integer, then the sequence will converge to 0. If n is a half-integer (i.e. n = 1/2, 3/2, 5/2, etc.), then the sequence will diverge.

2. What is the limit of the sequence b(n)= npicos(npi)?

The limit of this sequence is also dependent on the value of n. If n is an integer, then the limit will be 0. If n is a half-integer, then the sequence will not have a limit and will diverge.

3. How do we determine if a sequence is convergent or divergent?

A sequence is convergent if its terms approach a finite limit as n approaches infinity. A sequence is divergent if its terms do not approach a finite limit as n approaches infinity.

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

A convergent sequence approaches a finite limit as n approaches infinity, while a divergent sequence does not have a finite limit and its terms will either increase or decrease without bound.

5. Are there any other factors that can affect the convergence or divergence of a sequence?

Yes, there are other factors that can affect the convergence or divergence of a sequence. These include the form of the sequence, the values of n, and whether the sequence is an alternating or oscillating sequence.

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