Question

Could anyone please help me with this question?

Answer 1

It is converging.

Explanation:

To figure out if a sum of a series of terms converges or diverges we need to decide if the terms are getting bigger or smaller.

In this case the terms are summed for values of `k=1` to `k=oo`.

We need to work out if the value of the expression increases as `k` increases through this range of values.

Looking at the expression we can see that the numerator `sqrt(k^2+1)` is a square root, which has value

`sqrt2 = 1.414 (3dp)` at `k=1`.

It is then going to get closer and closer to k as k increases, so it is increasing at very nearly the same rate as k once k is large.

e.g. at `k=10, (sqrt(k^2+1))= sqrt101 = 10.050 (3dp)`

which is already pretty close to `k`

The denominator is 3 less than `k^3`, starting at -2 when `k=1` and increasing at close to `k^3` as `k` increases.

e.g. at `k=10, (k^3-3) = 997`

So we see that as `k->oo`, the expression `(sqrt(k^2+1))/(k^3-3) -> k/k^3 -> 1/oo`

So it is clearly converging.