Question

Determine if the following infinite series converges or diverges

Answer 1

Using limits, it is found that the infinite sequence converges, as the limit does not go to infinity.

How do we verify if a sequence converges of diverges?

Suppose an infinity sequence defined by:

$\sum_{k = 0}^{\infty} f(k)$

Then we have to calculate the following limit:

$\lim_{k \rightarrow \infty} f(k)$

If the limit goes to infinity, the sequence diverges, otherwise it converges.

In this problem, the function that defines the sequence is:

$f(k) = \frac{k^3}{k^4 + 10}$

Hence the limit is:

$\lim_{k \rightarrow \infty} f(k) = \lim_{k \rightarrow \infty} \frac{k^3}{k^4 + 10} = \lim_{k \rightarrow \infty} \frac{k^3}{k^4} = \lim_{k \rightarrow \infty} \frac{1}{k} = \frac{1}{\infty} = 0$

Hence, the infinite sequence converges, as the limit does not go to infinity.

More can be learned about convergent sequences at https://brainly.com/question/6635869

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Answer 2

The series diverges by the comparison test.

We have for large enough $k$,

$\displaystyle \frac{k^3}{k^4+10} \approx \frac{k^3}{k^4} = \frac1k$

so that

$\displaystyle \sum_{k=0}^\infty \frac{k^3}{k^4+10} = \frac1{10} + \sum_{k=1}^\infty \frac{k^3}{k^4+10} \approx \frac1{10} + \sum_{k=1}^\infty \frac1k$

and the latter sum is the divergent harmonic series.