MHB Converging Series: Tests & Tips for Finding Solutions

[marky1]

Hi,

I would like to as you you help please with finding whether the following three series converge.

\sum_{1}^{\infty} (-1)^kk³(5+k)^-2k

$$\sum_{k=1}^\infty(-1)^kk^3(5+k)^{-2k}$$

\sum_{2}^{\infty} sin(Pi/2+kPi)/(k^0.5lnk)

$$\sum_{k=2}^\infty\frac{\sin\left(\frac{\pi}{2}+k\pi\right)}{\sqrt k\ln k}$$

\sum_{1}^{\infty} (ksin(1+³)/(k+lnk)

$$\sum_{k=1}^\infty\frac{k\sin(1+?)}{k+\ln k}$$

I would be very grateful should you like to give me some hint (e.g. which test I should use), please.

For instance, for the first one I have tried the AHS test, but failed in showing that the series decreases.

For the second and third ones, I have not been able to find the integrals for the integral test and the ratio test seemed not to work either. I'm quite desperate, honestly.

Many thanks for any pointer and help.

 

[Greg]

Hi marky and welcome to MHB! :D

Notice that I've edited your post to include your sums in proper $\LaTeX$ code. I've left the originals so if there are any discrepancies you can point them out. Also, the third sum contains an unmatched exponent, so clarification is needed.

It's probably best that you show your work so that we may point out any errors and give guidance where needed instead of merely posting the methods, which may differ from what you have already learned and/or deduced.

Thanks.