Hi vbrasic:vbrasic said:I'm having trouble showing why the limit of this must be greater than one.
vbrasic said:Homework Statement
Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent.
Homework Equations
We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent.
The Attempt at a Solution
Applying the ratio test, we find that $$\bigg|\frac{a_{n+1}}{a_n}\bigg|=\frac{n+1}{z}.$$ I'm having trouble showing why the limit of this must be greater than one.
vbrasic said:Added in the missing absoute value. I think the reason it must be greater than one in the limit is because for any complex number, we may write it as ##re^{i\phi},## with magnitude ##r##. Given then that ##r## is finite, we have that the limit tends to ##\infty## because of the ##n## in the numerator. Does that sound okay?
When a series is divergent, it means that the sum of its terms is infinite or does not converge to a finite value. In other words, the series does not have a finite sum and its terms continue to increase without bound.
There are several techniques that can be used to show that a series is divergent. One common method is to use the comparison test, where the divergent series is compared to another known divergent series. Another approach is to use the divergence test, where the limit of the series' terms is taken and evaluated. If the limit is non-zero, then the series is divergent.
No, a series can only be either convergent or divergent. If a series is convergent, it means that its terms sum up to a finite value, while a divergent series has an infinite or non-converging sum. It is not possible for a series to have both of these characteristics.
Yes, it is possible for a series to have terms that approach zero, but still be divergent. This is because the terms may decrease at a very slow rate, causing the sum to continue increasing without bound.
Yes, a series can still be divergent even if its terms alternate between positive and negative. In this case, the series can be evaluated using the alternating series test, which considers the alternating behavior of the terms. If the series fails this test, it is divergent.