- #1
kskiraly
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Homework Statement
sum of tan(1/n)/(1+n) for n=1 to infinity
Homework Equations
The Attempt at a Solution
I tried using the ratio test and the comparison/limit comparison tests but can't think on anything to compare it to.
mybsaccownt said:I'm far from a math wiz, but here's how I see it:
tan(x) will always be between -1 and 1,
mybsaccownt said:and you know that 1/n converges...so by comparison, tan(1/n)/(1+n) must also converge
Convergence and divergence refer to the behavior of a series, which is a sum of an infinite sequence of numbers. A convergent series is one where the terms of the sequence approach a finite limit as the number of terms increases. Conversely, a divergent series is one where the terms of the sequence do not approach a finite limit, but instead grow larger and larger.
To determine the convergence or divergence of a series, you can use various tests such as the geometric series test, the comparison test, or the ratio test. These tests evaluate the behavior of the terms in the series and can help determine if the series will converge or diverge.
A convergent series has a finite sum, which means that the terms in the series add up to a specific value. This can have practical applications in fields such as finance and engineering. A divergent series, on the other hand, does not have a finite sum and can be used to represent quantities that grow without bound, such as population growth or radioactive decay.
No, a series can either converge or diverge, but not both. If a series converges, it cannot diverge, and vice versa. However, it is possible for a series to be neither convergent nor divergent, in which case it is said to be oscillatory.
The behavior of the terms in a series is crucial in determining its convergence or divergence. For example, if the terms in a series decrease in size and approach zero, the series is more likely to converge. On the other hand, if the terms in a series grow without bound, the series is more likely to diverge. Additionally, the rate at which the terms grow or decrease can also impact the convergence or divergence of a series.