how does this series converge?

Question

i need to prove that the series n^2 / ((n^4) +1) converges

how do i go about doing this? should i reduce the function first?? And what function do i compare it to?
NOTE: I must use the comparison test.

Answer 1

Note that:

n^2 / (n^4 + 1) = 1 / (n^2 + 1/n^2)

I just divided the numerator and denominator by n^2.

Now, for every n >= 1,

1 / (n^2 + 1/n^2) < 1/n^2.

But the series 1/n^2 converges (by the p-test), so therefore the original series must converge as well.

Answer 2

Note that 0 < n^2 / ((n^4) +1) < n^2 / n^4 = 1/(n^2)

Answer 3

You compare it to 1/n^2. You establish the convergence for n^2 first. Then build it up mathematically on both sides to see what the original converges to.

Answer 4

Compare it to 1/n^2, for large n.

Answer 5

better yet...

use limit comparison...
the function is 1/n^2

that is a hyperharmonic or a p-series which is convergent...
the series you have is thus convergent...

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