Checking series for convergence and limit

Question

Let $(x_k)_{k\in N}$ $\subset \mathbb{R^4} $.

Then there's this series, which I have to check for convergence and its limit.

I think that $(-1)^k * k$ diverges, because of the geometric series, which is saying that if $|q|^n$ $\geq 1$, the series diverges.

Now for the second part we have $(-1)^k * (1/k)$, which converges, because $1/k$ converges towards 0.

$(-1)^k * k^{300}$ diverges too, for the same reason like the first series.

$arctan (k) = \pi/2$, because it does not tend to 0 as k tends to infinity the divergence test tells us that the infinite series diverges.

I don't know if that is correct at all...

Answer 1

It just suffices to observe that since $(-1)^k\cdot k$ doesn't converges then $x_k$ doesn't converges too.

Moreover to conclude that $(-1)^k\cdot k$ doesn't converges we don't need geometric series but it suffices to observe that

• $(-1)^k\cdot k \to +\infty$ for $k$ even
• $(-1)^k\cdot k \to -\infty$ for $k$ odd

Answer 2

Are you finding the convergence of series or sequences?

If you are finding the convergence of series, your results are correct but your reasoning is not quite right.

For the second series with general term $(-1)^k /k $ you have argued that it is convergent because limit of $(-1)^k /k $ is $0$. Yes it is convergent because it is an alternating series with limit of the terms equal zero and the absolute values are decreasing. just the general term goes to zero is not enough for the series to converge.

Also on the last one you are using divergence test for series correctly but you you need to word it better. Over all you did fine but the notation and wording needs some improvement.