Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test

Question

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I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks

P.S. if my handwriting threw you off the original equation is $$\sum_{n=0}^{\infty} (-1)^n \cdot \frac{n^4}{n^3 + 1}.$$

Answer 1

There's nothing wrong with your calculations. You did well to find out the value of $b_n=x$, but it actually diverges, so when you compute the limit of your series over $b_n$, finding it to be 1 actually means that it diverges rather than converges.

Answer 2

Is one of your "diverges" supposed to be "converges"?

Your sum is essentially $\sum_{n=1}^{\infty} (-1)^nn $ or $-1+2-3+4-5+6...$ with partial sums $-1,1,-2,2,-3,3,...$. This diverges.

In what sense can the series converge?

If you are looking at Cesaro sums, the even ones are zero and the odd ones go to -1/2, so these do not converge.