Convergence of an Infinite Series ?

Question

I need to figure out if the infinite series of n^2*(-1)^n converges or diverges. It seems like it would diverge since n^2 is always increasing but I don't see how I can prove it with any of the tests that I know of since the alternating series test fails.

Answer 1

In the limit, not only is the value (positive or negative) of each term constantly increasing, but the *difference* between each respective terms increases. As n → ∞, the series doesn't converge, but *oscillates* between +∞ and -∞.

An alternative way of writing this series might be

Σ { (2n)² - (2n-1)² };

which makes it a little easier to see that the series diverges in the limit.

~W.O.M.B.A.T.

Answer 2

so some distance as #2 is going, you will compute the fallacious quintessential int(a million, infinity) x^3 * 3^{-x} dx. To compute this demands integration by way of areas some circumstances, even with the undeniable fact that that's in reality ordinary. If a = ln(3), then i've got self belief the quintessential evaluates to a/3 + a^2 + 2a^3 + 2a^4. even with the undeniable fact that, that's no longer so significant. All that concerns is that it converges. end: sum(a million, infinity) n^3/3^n <= int(a million,infinity) x^3/3^x dx < infinity and so the sum converges.

Answer 3

you can probably do it by the ratio test as stated by previous person, but i don't see why you can't say it is absolutely divergent, since,

abs{n^2*(-1)^n} = abs{n^2}

and n^2 diverges as you stated, so the series also diverges.

Answer 4

Use the limit test(or also called nth term test).