Question

Does the series converge or diverge?

sqrt(n^4+1)/5n^2

This one i actually need help on...what test do I use for this one? @LolWolf

Answer 1

Try using a direct comparison.

Here:
\[
\frac{\sqrt{n^4+1}}{5n^2}=\frac{1}{5}\sqrt{\frac{n^4}{n^4}+\frac{1}{n^4}}>\;?
\]

Answer 2

p>1 converges

Answer 3

i dont see how you got the fractions under the sqrt though

Answer 4

jk i see it now

Answer 5

sorry its been a long night!

Answer 6

No, it doesn't quite converge.

Note that:\[
\sqrt{\frac{n^4}{n^4}+\frac{1}{n^4}}=\sqrt{1+\frac{1}{n^4}}>1
\]
And, clearly:
\[
1+1+1+1+1+1+\cdots=\infty
\]

Answer 7

but since its larger than 1, because of the p-series it converges? I'm sorry im not quite seeing it

Answer 8

Remember that the p-series refers to the exponent of the inverse:
\[
\sum_n\frac{1}{n^p}
\]This is not an exponent, this is the actual number:
\[
\sum_n\sqrt{1+\frac{1}{n^4}}>\sum_n 1=\infty
\]Sum 1 infinitely many times and you should have infinity.

Answer 9

oooh okay! i get it now...thank you:)