Which series will converge? Pictures are attached, please explain how you got your answer.

Question

Which series will converge?  Pictures are attached, please explain how you got your answer.

anonymous · Answer

A.

anonymous · Answer

B.

anonymous · Answer

C.

anonymous · Answer

D.

anonymous · Answer

I think it could be C, but I'm not sure...

anonymous · Answer

@Directrix could you please help me with this?

anonymous · Answer

I really don't get it, please help.

anonymous · Answer

@satellite73  could you please help me with this?

anonymous · Answer

A will converge as well l

anonymous · Answer

in fact, i think that is the only one that converges from the ones you posted

anonymous · Answer

that's great, how did you figure it out?

anonymous · Answer

$$\sum\frac{1}{2}n=\frac{1}{2}\sum n=\frac{1}{2}(1+2+3+4+...$$ which pretty clearly does not converge, as the numbers get bigger and bigger

anonymous · Answer

$$\sum n^2$$ is even worse
you get 
$$1+4+9+16+25+...$$

anonymous · Answer

Right,

anonymous · Answer

the $\frac{1}{8}$ doesn't change anything 
still the numbers get bigger and bigger in $\sum\frac{1}{8}n^2$

anonymous · Answer

only 
$$\sum_{n=1}^{\infty}\frac{1}{n^2}$$ will converge, because the degree of the denominator is 2 more than the degree of the numerator

anonymous · Answer

@satellite73 Thank you so much!  You're a life saver!

anonymous · Answer

yw