1. convergence and Divergence problem

Please give me the help to find out the divergence or convergence of attached picture here:

Thanks

2. Originally Posted by dhammikai
Please give me the help to find out the divergence or convergence of attached picture here:

Thanks
Why don't they all converge?

3. To expand on Drexel's post a little, you have all the sums from 1 to $\alpha$. These are finite sums, so they all converge.

4. I think all the sums from 1 to $\infty$.

problem (a)(ii)
using the ratio test,
$\rho=\lim\limits_{n\longrightarrow\infty}\frac{a_{ n+1}}{a_{n}}=\lim
\limits_{n\longrightarrow\infty}\frac{4}{n+1}=0<1$

hence, this series converge.

5. I would consider that very ambiguous. What is shown as an upper limit is clearly " $\alpha$", not " $\infty$". But without a definition of $\alpha$, the problems simply don't make sense!