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quasar987 said:Are you saying it converges, or diverge?
quasar987 said:Based on what you said, I assume that you already understood that the series can be rewritten as
[tex]\sum_{n=1}^{\infty}2(-1)^{n}[/tex]
??
I don't see which test can be used on this. But you can fall back on the very definition of convergence: A series converge if the sequence of the partial sums converge. But if you find two subsequences that converge to different values, then the sequence itself diverges. Can you find those subsequences?
quasar987 said:Ok, here is how.
[tex]\frac{(-2)^{n+1}}{2^n}=\frac{(-2)(-2)^n}{2^n}=(-2)\left(\frac{(-2)}{2}\right)^n=(-2)(-1)^n=2(-1)^{n+1}[/tex]
Thus,
[tex]\sum_{n=0}^{\infty}\frac{(-2)^{n+1}}{2^n}=\sum_{n=0}^{\infty}2(-1)^{n+1}=\sum_{n=1}^{\infty}2(-1)^{n}[/tex]
chesshaha said:Thank you very much, this helps alot.
So the series converge, the sum is either 0 or -2, depends if it's even or odd, right?
Dick said:A series cannot converge to two limits. That sort of behavior is called 'divergent'.
Gib Z said:Ahh I think a more appropriate word would have been is oscillating =]
A series is said to converge if the sum of its terms approaches a finite value as the number of terms approaches infinity. In simpler terms, the terms of the series eventually add up to a specific number instead of getting larger and larger.
There are several tests that can be used to determine the convergence or divergence of a series, such as the ratio test, root test, and comparison test. These tests involve analyzing the behavior of the terms of the series to determine if they approach a finite value or if they diverge to infinity.
Knowing if a series converges or diverges is important in many areas of mathematics and science, as it allows for the accurate calculation of sums and the prediction of future values. It is also important in understanding the behavior of functions and in the development of mathematical theories.
Yes, a series can converge to different values depending on the specific terms of the series and the method used to determine convergence. For example, a series may converge to one value when using the ratio test, but may converge to a different value when using the root test.
A series is said to diverge if the sum of its terms does not approach a finite value as the number of terms approaches infinity. This can occur if the terms of the series get increasingly larger or if they oscillate between positive and negative values without approaching a specific value.