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pliu123123
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[itex]\sum sin k[/itex]
and k is from 1 to infinity,thx
and k is from 1 to infinity,thx
pliu123123 said:[itex]\sum sin k[/itex]
and k is from 1 to infinity,thx
dimension10 said:Nope. Its divergent. From 1 to 2pi, it is x. For 1 to infinty it is infinity x=infinity.
Erm, are you sure?Mute said:A sum that doesn't converge doesn't necessarily diverge.
HallsofIvy said:I could have sworn that the definition of "diverge" was "doen't converge"!
Which does NOT necessarily mean "diverge to infinity".
[tex]\sum_{n=0}^\infty (-1)^n[/tex]= 1- 1+ 1- 1+ 1- 1+...
has partial sums, 1, 0, 1, 0, 1, 0, ... which does not "diverge to infinity" but does "diverge".
Mute said:I stand corrected, then! I've always considered "diverge" to be short for "diverges to [itex]\pm \infty[/itex]" and something with partial sums which have no limit simply had no limit.
dimension10 said:Ok, it is an oscillating series, at least.
micromass said:Yes it is, but proving that might not be so obvious.
micromass said:For example, the series
[tex]\sum\sin(\pi k)[/tex]
does converge. I always ask this as exam question, only the ones who know what they're doing answer this correctly.
dimension10 said:At every pi, the sum becomes 0. infinity+pi/2=infinity. So infinity could just be a multiple of 2pi but it could have a modulus of pi/2. sin(pi/2)=1. So is this why the series is oscillating?
Does it converge to 0? because sin(pi*k)=0 (when k is an integer)
micromass said:Huh? First of all, you take sin(k) for integers k, so k never becomes pi (or an integer multiple of pi). So the sum never becomes zero. The rest of your post doesn't make much sense to me
The concept of convergence in mathematics refers to the idea that a sequence or series of numbers approaches a definite value or limit as more terms are added. In other words, as the terms of the sequence or series get closer and closer together, they eventually reach a fixed value.
There are several methods for determining the convergence or divergence of a series, including the ratio test, integral test, and comparison test. These methods involve comparing the series to known convergent or divergent series or using mathematical calculations to determine the behavior of the series.
The series sum (sin k) is known as the sine series and it is a divergent series. This can be shown using the limit comparison test or by graphing the series and observing that it does not approach a definite value as more terms are added.
Yes, there are some cases where a divergent series can be considered convergent. One example is the Grandi's series, which is the series 1 - 1 + 1 - 1 + 1 - 1 + ... This series does not have a definite value, but it can be assigned a value of 1/2 using a mathematical technique called Cesàro summation.
The concept of convergence is widely used in various fields of science and engineering, including physics, economics, and computer science. In physics, the convergence of series is used to calculate quantities such as electric potential and magnetic field. In economics, it is used to determine the value of investments over time. In computer science, the convergence of series is used in algorithms for numerical analysis and data compression.