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incubusfan723
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Homework Statement
I was given the series sin 1 + sin 2 + sin 3 + sin 4 + ... and I must show that it diverges. Can anyone point me in the right direction as to how to go about doing this?
xcvxcvvc said:If the parts of a series never reach zero, wouldn't you keep adding chunks, so the sum never settles down(never converges - it diverges)?
chroot said:
berkeman said:Looks like the first rule only applies for summands > 0. Good link, though. Time for some reading...
chroot said:No, berkeman. The summand in this case is sin(n), and the limit of sin(n) as n goes to infinity is undefined. Therefore, the series does not converge.
- Warren
chroot said:
berkeman said:Note that the range of sin(x) is -1 to 1. So if you picked the arguments to the sin(x) function correctly, you could get the sum to not diverge.
The hard part of this qestion is that the x values are not an even fraction of PI...
Mark44 said:What they refer to in the wiki article as "Limit of the summand" is similar to one that appears in some calculus texts, the Nth Term Test for Divergence. I.e., in a series [itex]\sum a_n[/itex], if lim an is not zero, the series diverges.
A common mistake that students make happens when they find that lim an = 0, and conclude that [itex]\sum a_n[/itex] converges.
berkeman said:The sum can oscillate without diverging.
I agree completely. A series doesn't have to have sums that become ever larger or ever more negative to diverge.chroot said:At least in the mathematical sense, oscillation is a form of divergence, because the series sum never settles down to any specific value. The nth term test is all that's needed to show that this series diverges. I've even heard this called the "hurdle test" by math teachers, because if it doesn't pass this hurdle, it cannot converge, and there's no need to check anything else.
- Warren
A divergent series is a sequence of terms that does not have a finite sum, meaning that as the number of terms increases, the sum of the series also increases without bound.
To determine if a series diverges or converges, you can use various convergence tests such as the comparison test, ratio test, or the integral test. These tests involve comparing the series to a known convergent or divergent series or evaluating the series using integrals.
The sum of an infinite series is the value that the series approaches as the number of terms increases without bound. In the case of a divergent series, the sum is said to be infinity.
This series diverges because the sequence of terms does not approach a finite value as the number of terms increases. The values of the sine function oscillate between -1 and 1, so as more terms are added, the sum will continue to increase without bound.
No, a divergent series cannot have a finite sum. By definition, a divergent series does not have a finite sum and the sum will continue to increase without bound as more terms are added.