Question

determine whether the series converges or diverges:
(-1)^n n^2/(5^n)

Answer 1

\[(-1)^n \frac{ n^2 }{ 5^n }\]

Answer 2

i believe it's divergent

Answer 3

I looks convergent. To prove this, you have to prove that the absolute value of the number doesn't get larger and larger.
I would do it this way:\[|(-1)^n \frac{n^2}{5^n}|=\frac{n^2}{5^n}< \frac{4^n}{5^n}=\left( \frac{4}{5} \right)^n \rightarrow 0 \] So the limit is 0. The series converges.

Some would say this could even be done much simpler: it is well-known that exponential functions increase faster than power functions, so the denominator "wins" and the limit is 0. In fact, the argument in my calculations is the same: \(n^2 < 4^n\). The exponential sequence increases faster than the power sequence.

I don't know what you are supposed to do, but these are two ways to approach it.

Answer 4

try using ratio test

Answer 5

there are couple of other test ... the above mentioned test is somewhat close to Leibniz test.

Answer 6

it works!

Answer 7

thanks to you both! how can i award both metals?

Answer 8

YW!
You can't :(